Chapter 14 Mathematical Reasoning

A compound statement is a statement formed by combining two or more simple statements using logical connectives like 'and', 'or', 'if...then', 'if and only if', 'neither...nor', etc.

(i) “2 is both an even number and a prime number”

This statement is a combination of two simple statements:

Statement 1: "2 is an even number"

Statement 2: "2 is a prime number"

These are connected by the word "and". Thus, it is a compound statement.

(ii) “9 is neither an even number nor a prime number”

This statement can be rewritten as "9 is not an even number and 9 is not a prime number".

This is a combination of two simple statements (negations of simpler statements):

Statement 1: "9 is not an even number"

Statement 2: "9 is not a prime number"

These are connected by the word "and" (implied by "neither...nor"). Thus, it is a compound statement.

(iii) “Ram and Rahim are friends”

This statement is a single proposition about two subjects sharing the property of being friends. The word "and" here connects the subjects "Ram" and "Rahim" within a single simple predicate, rather than connecting two distinct simple statements.

Thus, it is not a compound statement.

Therefore, the compound statements among the given options are (i) and (ii).

A compound statement is formed by combining simple statements using logical connectives. We need to identify these simple statements (component statements) and the connective used.

(a) “It is raining or the sun is shining.”

The connective used is "or".

The component statements are:

Statement 1: "It is raining."

Statement 2: "The sun is shining."

(b) “2 is a positive number or a negative number.”

The connective used is "or".

The component statements are:

Statement 1: "2 is a positive number."

Statement 2: "2 is a negative number."

To translate statements into symbolic form, we represent the simple component statements with letters (usually p, q, r, etc.) and the logical connectives with symbols (e.g., $\wedge$ for "and", $\vee$ for "or").

(i) “2 and 3 are prime numbers”

This statement combines two simple statements using the connective "and".

Let p: "2 is a prime number"

Let q: "3 is a prime number"

The connective is "and".

In symbolic form, the statement is: $p \wedge q$.

(ii) “Tigers are found in Gir forest or Rajaji national park.”

This statement combines two simple statements using the connective "or".

Let p: "Tigers are found in Gir forest."

Let q: "Tigers are found in Rajaji national park."

The connective is "or".

In symbolic form, the statement is: $p \vee q$.

To find the truth value of a compound statement, we determine the truth values of its component statements and apply the rules for the logical connective used.

(i) “9 is an even integer or 9 + 1 is even.”

This is a compound statement with the connective "or".

Let p: "9 is an even integer." The truth value of p is False ($F$), as 9 is an odd integer.

Let q: "9 + 1 is even." $9 + 1 = 10$. The truth value of q is True ($T$), as 10 is an even integer.

The compound statement is $p \vee q$. Since q is true, the statement $p \vee q$ is True.

(ii) “2 + 4 = 6 or 2 + 4 = 7”

This is a compound statement with the connective "or".

Let p: "$2 + 4 = 6$." The truth value of p is True ($T$).

Let q: "$2 + 4 = 7$." The truth value of q is False ($F$).

The compound statement is $p \vee q$. Since p is true, the statement $p \vee q$ is True.

(iii) “Delhi is the capital of India and Islamabad is the capital of Pakistan.”

This is a compound statement with the connective "and".

Let p: "Delhi is the capital of India." The truth value of p is True ($T$).

Let q: "Islamabad is the capital of Pakistan." The truth value of q is True ($T$).

The compound statement is $p \wedge q$. Since both p and q are true, the statement $p \wedge q$ is True.

(iv) “Every rectangle is a square and every square is a rectangle.”

This is a compound statement with the connective "and".

Let p: "Every rectangle is a square." The truth value of p is False ($F$), as a rectangle only needs to have four right angles, not equal sides.

Let q: "Every square is a rectangle." The truth value of q is True ($T$), as a square satisfies the definition of a rectangle (four right angles and opposite sides equal).

The compound statement is $p \wedge q$. Since p is false, the statement $p \wedge q$ is False.

(v) “The sun is a star or sun is a planet.”

This is a compound statement with the connective "or".

Let p: "The sun is a star." The truth value of p is True ($T$).

Let q: "Sun is a planet." The truth value of q is False ($F$).

The compound statement is $p \vee q$. Since p is true, the statement $p \vee q$ is True.

The negation of a statement reverses its truth value. If a statement is true, its negation is false, and vice versa.

The given statement is a universal statement claiming that a property holds for "everyone who lives in India".

Original Statement: "Everyone who lives in India is an Indian."

To negate a universal statement ("Everyone has property P"), we state that "There exists at least one person who does not have property P".

In this case, the property is "is an Indian" for those who "live in India".

So, the negation is: "There exists someone who lives in India and is not an Indian."

Negation of the statement: There exists a person who lives in India but is not an Indian.

The negation of a statement is a statement that is true when the original statement is false, and false when the original statement is true.

(a) p : “All triangles are equilateral triangles.”

This is a universal statement. The negation of "All A are B" is "There exists an A that is not B".

Negation of p: There exists a triangle that is not an equilateral triangle.

This can also be stated as: Some triangles are not equilateral triangles.

(b) q : “9 is a multiple of 4.”

This is a simple statement. The negation is simply denying the statement.

Negation of q: 9 is not a multiple of 4.

(c) r : “A triangle has four sides.”

This is a simple statement. The negation is simply denying the statement.

Negation of r: A triangle does not have four sides.

To find the negation of a compound statement involving "or" or "and", we use De Morgan's Laws:

Negation of "$p \vee q$" is "$\neg p \wedge \neg q$".

Negation of "$p \wedge q$" is "$\neg p \vee \neg q$".

Here, $\neg p$ and $\neg q$ are the negations of the simple statements p and q, respectively.

(i) “Suresh lives in Bhopal or he lives in Mumbai.”

Let p: "Suresh lives in Bhopal."

Let q: "Suresh lives in Mumbai."

The statement is of the form $p \vee q$.

Its negation is $\neg (p \vee q) \equiv \neg p \wedge \neg q$.

$\neg p$: "Suresh does not live in Bhopal."

$\neg q$: "Suresh does not live in Mumbai."

The negation in words is: Suresh does not live in Bhopal and he does not live in Mumbai.

Alternatively: Suresh neither lives in Bhopal nor lives in Mumbai.

(ii) “x + y = y + x and 29 is a prime number.”

Let p: "$x + y = y + x$."

Let q: "29 is a prime number."

The statement is of the form $p \wedge q$.

Its negation is $\neg (p \wedge q) \equiv \neg p \vee \neg q$.

$\neg p$: "$x + y \neq y + x$."

$\neg q$: "29 is not a prime number."

The negation in words is: $x + y \neq y + x$ or 29 is not a prime number.

A conditional statement is typically written in the form "If p, then q", where p is the antecedent (the condition) and q is the consequent (the result).

(i) “Mohan will be a good student if he studies hard.”

The statement can be rephrased to identify the condition and the outcome.

Condition: He studies hard.

Outcome: Mohan will be a good student.

Conditional statement: If Mohan studies hard, then he will be a good student.

(ii) “Ramesh will get dessert only if he eats his dinner.”

The phrase "only if" indicates a necessary condition. "A only if B" means that B is a necessary condition for A. This is equivalent to "If A, then B".

A: Ramesh will get dessert.

B: He eats his dinner.

Conditional statement: If Ramesh gets dessert, then he eats his dinner.

(iii) “When you sing, my ears hurt.”

The word "When" indicates a condition in this context.

Condition: You sing.

Outcome: My ears hurt.

Conditional statement: If you sing, then my ears hurt.

(iv) “A necessary condition for Indian team to win a cricket match is that the selection committee selects an all-rounder.”

"B is a necessary condition for A" means "If A, then B".

A: Indian team wins a cricket match.

B: The selection committee selects an all-rounder.

Conditional statement: If the Indian team wins a cricket match, then the selection committee selects an all-rounder.

(v) “A sufficient condition for Tara to visit New Delhi is that she goes to the Rashtrapati Bhawan.”

"B is a sufficient condition for A" means "If B, then A".

A: Tara visits New Delhi.

B: She goes to the Rashtrapati Bhawan.

Conditional statement: If Tara goes to the Rashtrapati Bhawan, then she visits New Delhi.

The symbolic statement $p \to q$ represents a conditional statement. The symbol $\to$ is read as "if ... then ...".

Given statements:

p: "It is raining today"

q: "$2 + 3 > 4$"

The statement $p \to q$ means "If p, then q".

Translating this into English, we get: If it is raining today, then $2 + 3 > 4$.

To translate this statement into symbolic form, we first identify the simple statements and the connectives used. This statement is in the form "If ..., then ...", which is a conditional statement.

The antecedent (the part after "If" but before "then") is "x = 7 and y = 4". This itself is a compound statement using the connective "and".

The consequent (the part after "then") is "x + y = 11".

Let's define the simple statements:

p: "$x = 7$"

q: "$y = 4$"

r: "$x + y = 11$"

The antecedent "x = 7 and y = 4" can be written symbolically as $p \wedge q$.

The consequent is $r$.

The conditional statement "If $(p \wedge q)$, then $r$" is written symbolically using the $\to$ connective.

The symbolic form of the statement is: $(p \wedge q) \to r$.

A biconditional statement is formed by combining two simple statements p and q using the connective "if and only if". It is denoted by $p \leftrightarrow q$. The statement $p \leftrightarrow q$ is true if and only if p and q have the same truth value (both true or both false).

Given statements:

p: "Today is 14^th of August"

q: "Tomorrow is Independence day"

To form the biconditional statement $p \leftrightarrow q$, we combine p and q with "if and only if".

The biconditional statement is: Today is 14^th of August if and only if tomorrow is Independence day.

A biconditional statement of the form "p if and only if q" is translated into symbolic form as $p \leftrightarrow q$. We need to identify the simple statements p and q.

Let p be the statement: "ABC is an equilateral triangle"

Let q be the statement: "Its each interior angle is 60°" (referring to triangle ABC)

The connective "if and only if" is represented by the symbol $\leftrightarrow$.

The symbolic form of the biconditional statement is: $p \leftrightarrow q$.

A quantifier is a phrase that indicates how many objects a statement refers to. Common quantifiers are "There exists" (existential quantifier, $\exists$) and "For all" (universal quantifier, $\forall$). The negation of a statement with a quantifier reverses the quantifier and negates the predicate.

(i) “There exists a number which is equal to its square.”

The quantifier is: There exists ($\exists$).

Let P(x) be the predicate "x is equal to its square" ($x = x^2$). The statement is "$\exists x, P(x)$".

The negation of "There exists an x such that P(x)" is "For all x, not P(x)".

Negation: For all numbers, it is not equal to its square.

Alternatively: Every number is not equal to its square.

(ii) “For all even integers x, x² is also even.”

The quantifier is: For all even integers x ($\forall$ x $\in$ {even integers}).

Let P(x) be the predicate "$x^2$ is also even". The statement is "$\forall x, P(x)$".

The negation of "For all x, P(x)" is "There exists an x such that not P(x)".

Negation: There exists an even integer x such that x² is not even.

(iii) “There exists a number which is a multiple of 6 and 9.”

The quantifier is: There exists ($\exists$).

Let P(x) be the predicate "x is a multiple of 6" and Q(x) be the predicate "x is a multiple of 9". The statement is "$\exists x, (P(x) \wedge Q(x))$".

The negation of "There exists an x such that (P(x) and Q(x))" is "For all x, not (P(x) and Q(x))".

Using De Morgan's Law, $\neg (P(x) \wedge Q(x)) \equiv \neg P(x) \vee \neg Q(x)$.

Negation: For all numbers, it is not a multiple of 6 or it is not a multiple of 9.

Alternatively: No number is a multiple of both 6 and 9.

We need to show that the conditional statement "if $x = y$, then $2x + a = 2y + a$" is true for any real numbers x, y and any integer a. To prove a conditional statement $P \to Q$, we assume the antecedent P is true and show that the consequent Q must also be true.

Given:

x and y are real numbers ($x, y \in \mathbb{R}$).

a is an integer ($a \in \mathbb{Z}$).

Assume the antecedent is true: $x = y$.

To Prove:

$2x + a = 2y + a$

Proof:

We are given that $x = y$.

Since x and y are real numbers, and $x = y$, we can multiply both sides of the equality by the real number 2. The property of equality states that if $A = B$, then $c \cdot A = c \cdot B$ for any number c.

Now, we can add the same integer 'a' to both sides of the equality. The property of equality states that if $A = B$, then $A + c = B + c$ for any number c. Since $a \in \mathbb{Z}$, $a$ is also a real number.

We have successfully derived the consequent ($2x + a = 2y + a$) from the assumption that $x = y$.

Thus, the statement "For any real numbers x, y, if x = y, then 2x + a = 2y + a when a ∈ Z" is True.

To check the validity of a compound statement, we determine its truth value. A statement is considered valid if it is true.

(i) r : “100 is a multiple of 4 and 5.”

This is a compound statement with the connective "and".

Let p: "100 is a multiple of 4." This is True ($100 = 4 \times 25$).

Let q: "100 is a multiple of 5." This is True ($100 = 5 \times 20$).

The compound statement is $p \wedge q$. For an "and" statement to be true, both component statements must be true.

Since both p and q are true, the statement $p \wedge q$ is True.

Therefore, the statement r is Valid.

(ii) s : “60 is a multiple of 3 or 5.”

This is a compound statement with the connective "or".

Let p: "60 is a multiple of 3." This is True ($60 = 3 \times 20$).

Let q: "60 is a multiple of 5." This is True ($60 = 5 \times 12$).

The compound statement is $p \vee q$. For an "or" statement to be true, at least one of the component statements must be true.

Since p is true (and q is also true), the statement $p \vee q$ is True.

Therefore, the statement s is Valid.

A statement is a declarative sentence that is either true or false, but not both.

Let's examine each option:

(A) "Roses are black." This is a declarative sentence. It is false, but it has a definite truth value. Thus, it is a statement.

(B) "Mind your own business." This is an imperative sentence (a command). It is neither true nor false.

(C) "Be punctual." This is an imperative sentence (a command). It is neither true nor false.

(D) "Do not tell lies." This is an imperative sentence (a command). It is neither true nor false.

Only option (A) is a declarative sentence that can be judged as true or false.

The correct answer is (A) Roses are black.

To find the negation of a compound statement connected by "and", we use De Morgan's Law.

Let p be the statement: "It is raining."

Let q be the statement: "Weather is cold."

The given statement is of the form $p \wedge q$.

The negation of $p \wedge q$ is given by De Morgan's Law:

$\neg (p \wedge q) \equiv \neg p \vee \neg q$

The negation of p is $\neg p$: "It is not raining."

The negation of q is $\neg q$: "Weather is not cold."

Combining $\neg p$ and $\neg q$ with the connective "or", we get the negation of the original statement.

Negation in words: It is not raining or weather is not cold.

Comparing this with the given options, we find that option (C) matches our result.

The correct answer is (C) It is not raining or weather is not cold.

For a conditional statement of the form "If p, then q" (symbolically $p \to q$), the converse is formed by interchanging the antecedent (p) and the consequent (q). The converse is the statement "If q, then p" (symbolically $q \to p$).

Let p be the statement: "Billu secure good marks."

Let q be the statement: "he will get a bicycle."

The given statement is "If p, then q".

The converse statement is "If q, then p".

Substituting the statements for p and q, the converse is: If he will get a bicycle, then Billu secure good marks.

This can be rephrased as: If Billu will get a bicycle, then he will secure good marks.

Comparing this with the given options, we find that option (B) matches our derived converse statement.

The correct answer is (B) If Billu will get a bicycle, then he will secure good marks.

A sentence is called a mathematical statement if it is either true or false but not both.

(i) A triangle has three sides.

This is a declarative sentence and is always true. Therefore, it is a statement.

(ii) 0 is a complex number.

This is a declarative sentence. A complex number is of the form $a + bi$, where $a$ and $b$ are real numbers. $0$ can be written as $0 + 0i$, where $a=0$ and $b=0$ are real numbers. Thus, it is always true. Therefore, it is a statement.

(iii) Sky is red.

This is a declarative sentence. It is always false. Therefore, it is a statement.

(iv) Every set is an infinite set.

This is a declarative sentence. This is false because there exist finite sets (e.g., the set $\{1, 2\}$). Therefore, it is a statement.

(v) $15 + 8 > 23$.

This is a declarative sentence. We have $15 + 8 = 23$. The sentence states $23 > 23$, which is false. Therefore, it is a statement.

(vi) $y + 9 = 7$.

This is a sentence whose truth value depends on the value of $y$. It is true for $y = -2$ and false for any other value of $y$. Since its truth value is not fixed as either true or false, it is not a statement.

(vii) Where is your bag?

This is an interrogative sentence (a question). Questions are not statements because they cannot be judged as true or false. Therefore, it is not a statement.

(viii) Every square is a rectangle.

This is a declarative sentence. A square satisfies the properties of a rectangle (opposite sides are parallel and equal, all angles are $90^\circ$). This sentence is always true. Therefore, it is a statement.

(ix) Sum of opposite angles of a cyclic quadrilateral is $180^\circ$.

This is a declarative sentence stating a known geometrical theorem. This sentence is always true. Therefore, it is a statement.

(x) $\sin^2 x + \cos^2 x = 0$

This is a declarative sentence. From the trigonometric identity, we know that $\sin^2 x + \cos^2 x = 1$ for all real values of $x$. Thus, the sentence $\sin^2 x + \cos^2 x = 0$ is always false. Therefore, it is a statement.

A compound statement is a statement that is formed by combining two or more simple statements. The simple statements that make up the compound statement are called its component statements.

(i) Number 7 is prime and odd.

This is a compound statement with the connective "and". The component statements are:

Component 1: Number 7 is prime.

Component 2: Number 7 is odd.

(ii) Chennai is in India and is the capital of Tamil Nadu.

This is a compound statement with the connective "and". The component statements are:

Component 1: Chennai is in India.

Component 2: Chennai is the capital of Tamil Nadu.

(iii) The number 100 is divisible by 3, 11 and 5.

This is a compound statement with the connective "and". The component statements are:

Component 1: The number 100 is divisible by 3.

Component 2: The number 100 is divisible by 11.

Component 3: The number 100 is divisible by 5.

(iv) Chandigarh is the capital of Haryana and U.P.

This is a compound statement with the connective "and". The component statements are:

Component 1: Chandigarh is the capital of Haryana.

Component 2: Chandigarh is the capital of U.P.

(v) $\sqrt{7}$ is a rational number or an irrational number.

This is a compound statement with the connective "or". The component statements are:

Component 1: $\sqrt{7}$ is a rational number.

Component 2: $\sqrt{7}$ is an irrational number.

(vi) 0 is less than every positive integer and every negative integer.

This is a compound statement with the connective "and". The component statements are:

Component 1: 0 is less than every positive integer.

Component 2: 0 is less than every negative integer.

(vii) Plants use sunlight, water and carbon dioxide for photosynthesis.

This is a compound statement with the connective "and". The component statements are:

Component 1: Plants use sunlight for photosynthesis.

Component 2: Plants use water for photosynthesis.

Component 3: Plants use carbon dioxide for photosynthesis.

(viii) Two lines in a plane either intersect at one point or they are parallel.

This is a compound statement with the connective "or". The component statements are:

Component 1: Two lines in a plane intersect at one point.

Component 2: Two lines in a plane are parallel.

(ix) A rectangle is a quadrilateral or a 5 - sided polygon.

This is a compound statement with the connective "or". The component statements are:

Component 1: A rectangle is a quadrilateral.

Component 2: A rectangle is a 5 - sided polygon.

To check if a compound statement is true or false, we first identify its component statements and determine their individual truth values. Then, we apply the rules for the given connective (like "and" or "or").

(i) 57 is divisible by 2 or 3.

Component statements:

Component 1 (P): 57 is divisible by 2.

Component 2 (Q): 57 is divisible by 3.

Truth values of component statements:

P: $57 \div 2$ gives a remainder of 1. So, 57 is not divisible by 2. This statement is false.

Q: $57 = 3 \times 19$. So, 57 is divisible by 3. This statement is true.

The compound statement is of the form "P or Q". For an "or" statement to be true, at least one of the components must be true. Since Q is true, the compound statement is true.

(ii) 24 is a multiple of 4 and 6.

Component statements:

Component 1 (P): 24 is a multiple of 4.

Component 2 (Q): 24 is a multiple of 6.

Truth values of component statements:

P: $24 = 4 \times 6$. So, 24 is a multiple of 4. This statement is true.

Q: $24 = 6 \times 4$. So, 24 is a multiple of 6. This statement is true.

The compound statement is of the form "P and Q". For an "and" statement to be true, both components must be true. Since both P and Q are true, the compound statement is true.

(iii) All living things have two eyes and two legs.

Component statements:

Component 1 (P): All living things have two eyes.

Component 2 (Q): All living things have two legs.

Truth values of component statements:

P: This statement is false (e.g., insects have multiple eyes, some animals have one eye or no eyes). This statement is false.

Q: This statement is false (e.g., snakes have no legs, insects have six or more legs, birds have two legs but animals like dogs and cats have four). This statement is false.

The compound statement is of the form "P and Q". For an "and" statement to be true, both components must be true. Since both P and Q are false, the compound statement is false.

(iv) 2 is an even number and a prime number.

Component statements:

Component 1 (P): 2 is an even number.

Component 2 (Q): 2 is a prime number.

Truth values of component statements:

P: An even number is divisible by 2. $2 \div 2 = 1$. So, 2 is an even number. This statement is true.

Q: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The divisors of 2 are 1 and 2. So, 2 is a prime number. This statement is true.

The compound statement is of the form "P and Q". For an "and" statement to be true, both components must be true. Since both P and Q are true, the compound statement is true.

The negation of a statement is formed by denying the statement. If a statement is denoted by $p$, its negation is denoted by $\sim p$ (read as "not $p$").

(i) The number 17 is prime.

Negation: The number 17 is not prime.

(ii) $2 + 7 = 6$.

Negation: $2 + 7 \neq 6$.

(iii) Violets are blue.

Negation: Violets are not blue.

(iv) $\sqrt{5}$ is a rational number.

Negation: $\sqrt{5}$ is not a rational number.

(v) 2 is not a prime number.

Negation: 2 is a prime number.

(vi) Every real number is an irrational number.

Negation: Not every real number is an irrational number. (Or, There exists at least one real number which is not an irrational number).

(vii) Cow has four legs.

Negation: Cow does not have four legs.

(viii) A leap year has 366 days.

Negation: A leap year does not have 366 days.

(ix) All similar triangles are congruent.

Negation: Not all similar triangles are congruent. (Or, There exists a similar triangle which is not congruent).

(x) Area of a circle is same as the perimeter of the circle.

Negation: Area of a circle is not same as the perimeter of the circle.

To translate statements into symbolic form, we identify the simple statements and the logical connectives (like "and", "or") used to combine them. We represent the simple statements with symbols (usually capital letters) and the connectives with logical symbols.

(i) Rahul passed in Hindi and English.

Let P: Rahul passed in Hindi.

Let Q: Rahul passed in English.

The connective is "and" ($\wedge$).

Symbolic form: $P \wedge Q$

(ii) x and y are even integers.

Let P: x is an even integer.

Let Q: y is an even integer.

The connective is "and" ($\wedge$).

Symbolic form: $P \wedge Q$

(iii) 2, 3 and 6 are factors of 12.

Let P: 2 is a factor of 12.

Let Q: 3 is a factor of 12.

Let R: 6 is a factor of 12.

The connective is "and" ($\wedge$).

Symbolic form: $P \wedge Q \wedge R$

(iv) Either x or x + 1 is an odd integer.

Let P: x is an odd integer.

Let Q: x + 1 is an odd integer.

The connective is "either ... or ..." ($\vee$).

Symbolic form: $P \vee Q$

(v) A number is either divisible by 2 or 3.

Let P: A number is divisible by 2.

Let Q: A number is divisible by 3.

The connective is "either ... or ..." ($\vee$).

Symbolic form: $P \vee Q$

(vi) Either $x = 2$ or $x = 3$ is a root of $3x^2 – x – 10 = 0$

Let P: $x = 2$ is a root of $3x^2 – x – 10 = 0$.

Let Q: $x = 3$ is a root of $3x^2 – x – 10 = 0$.

The connective is "either ... or ..." ($\vee$).

Symbolic form: $P \vee Q$

(vii) Students can take Hindi or English as an optional paper.

Let P: Students can take Hindi as an optional paper.

Let Q: Students can take English as an optional paper.

The connective is "or" ($\vee$).

Symbolic form: $P \vee Q$

To find the negation of a compound statement using "and" ($\wedge$) or "or" ($\vee$), we use De Morgan's Laws:

• Negation of "$P \wedge Q$" is "$\sim P \vee \sim Q$".
• Negation of "$P \vee Q$" is "$\sim P \wedge \sim Q$".

For statements involving "All", the negation often involves "Not all" or "There exists at least one... that is not...".

(i) All rational numbers are real and complex.

This statement can be rephrased as: For every number $x$, if $x$ is rational, then ($x$ is real and $x$ is complex).

The negation is: There exists a number $x$ such that $x$ is rational and it is not (real and complex).

Using De Morgan's law for the part "not (real and complex)", we get "not real or not complex".

Negation: There exists a rational number that is not real or is not complex.

Alternatively, more simply: Not all rational numbers are real and complex.

(ii) All real numbers are rationals or irrationals.

This statement can be rephrased as: For every number $x$, if $x$ is real, then ($x$ is rational or $x$ is irrational).

The negation is: There exists a number $x$ such that $x$ is real and it is not (rational or irrational).

Using De Morgan's law for the part "not (rational or irrational)", we get "not rational and not irrational".

Negation: There exists a real number that is neither rational nor irrational.

Alternatively, more simply: Not all real numbers are rationals or irrationals.

(iii) $x = 2$ and $x = 3$ are roots of the Quadratic equation $x^2 – 5x + 6 = 0$.

Let P: $x = 2$ is a root of $x^2 – 5x + 6 = 0$.

Let Q: $x = 3$ is a root of $x^2 – 5x + 6 = 0$.

The statement is $P \wedge Q$. The negation is $\sim(P \wedge Q) \equiv \sim P \vee \sim Q$.

Negation: $x = 2$ is not a root of the Quadratic equation $x^2 – 5x + 6 = 0$ or $x = 3$ is not a root of the Quadratic equation $x^2 – 5x + 6 = 0$.

(iv) A triangle has either 3-sides or 4-sides.

Let P: A triangle has 3 sides.

Let Q: A triangle has 4 sides.

The statement is $P \vee Q$. The negation is $\sim(P \vee Q) \equiv \sim P \wedge \sim Q$.

Negation: A triangle does not have 3 sides and does not have 4 sides.

(v) 35 is a prime number or a composite number.

Let P: 35 is a prime number.

Let Q: 35 is a composite number.

The statement is $P \vee Q$. The negation is $\sim(P \vee Q) \equiv \sim P \wedge \sim Q$.

Negation: 35 is not a prime number and 35 is not a composite number.

(vi) All prime integers are either even or odd.

This statement can be rephrased as: For every integer $n$, if $n$ is prime, then ($n$ is even or $n$ is odd).

The negation is: There exists an integer $n$ such that $n$ is prime and it is not (even or odd).

Using De Morgan's law for the part "not (even or odd)", we get "not even and not odd".

Negation: There exists a prime integer that is neither even nor odd.

Alternatively, more simply: Not all prime integers are either even or odd.

(vii) $|x|$ is equal to either $x$ or – $x$.

Let P: $|x| = x$.

Let Q: $|x| = -x$.

The statement is $P \vee Q$. The negation is $\sim(P \vee Q) \equiv \sim P \wedge \sim Q$.

Negation: $|x|$ is not equal to $x$ and $|x|$ is not equal to $-x$.

(viii) 6 is divisible by 2 and 3.

Let P: 6 is divisible by 2.

Let Q: 6 is divisible by 3.

The statement is $P \wedge Q$. The negation is $\sim(P \wedge Q) \equiv \sim P \vee \sim Q$.

Negation: 6 is not divisible by 2 or 6 is not divisible by 3.

A conditional statement is a statement that can be expressed in the form "If P, then Q", where P is the hypothesis and Q is the conclusion. This is symbolically represented as $P \to Q$.

(i) The square of an odd number is odd.

This statement means that if a number is odd, then its square is odd.

Conditional form: If a number is odd, then its square is odd.

(ii) You will get a sweet dish after the dinner.

This statement implies that the condition for getting a sweet dish is having dinner.

Conditional form: If you have the dinner, then you will get a sweet dish.

(iii) You will fail, if you will not study.

The "if" clause specifies the condition (hypothesis).

Conditional form: If you will not study, then you will fail.

(iv) The unit digit of an integer is 0 or 5 if it is divisible by 5.

The "if" clause specifies the condition (hypothesis).

Conditional form: If an integer is divisible by 5, then its unit digit is 0 or 5.

(v) The square of a prime number is not prime.

This statement means that if a number is prime, then its square is not prime.

Conditional form: If a number is prime, then its square is not prime.

(vi) $2b = a + c$, if a, b and c are in A.P.

The "if" clause specifies the condition (hypothesis).

Conditional form: If a, b and c are in A.P., then $2b = a + c$.

A biconditional statement $p \leftrightarrow q$ is read as "$p$ if and only if $q$". It is equivalent to the compound statement "$p$ implies $q$ and $q$ implies $p$".

(i) p : The unit digit of an integer is zero.

q : It is divisible by 5.

The biconditional statement $p \leftrightarrow q$ is formed by combining $p$ and $q$ with "if and only if".

Biconditional statement: The unit digit of an integer is zero if and only if it is divisible by 5.

(ii) p : A natural number n is odd.

q : Natural number n is not divisible by 2.

The biconditional statement $p \leftrightarrow q$ is formed by combining $p$ and $q$ with "if and only if".

Biconditional statement: A natural number $n$ is odd if and only if natural number $n$ is not divisible by 2.

(iii) p : A triangle is an equilateral triangle.

q : All three sides of a triangle are equal.

The biconditional statement $p \leftrightarrow q$ is formed by combining $p$ and $q$ with "if and only if".

Biconditional statement: A triangle is an equilateral triangle if and only if all three sides of a triangle are equal.

The contrapositive of a conditional statement "If P, then Q" is the statement "If not Q, then not P". Symbolically, the contrapositive of $P \to Q$ is $\sim Q \to \sim P$.

(i) If $x = y$ and $y = 3$, then $x = 3$.

P: $x = y$ and $y = 3$

Q: $x = 3$

$\sim$P: $x \neq y$ or $y \neq 3$

$\sim$Q: $x \neq 3$

Contrapositive: If $x \neq 3$, then $x \neq y$ or $y \neq 3$.

(ii) If $n$ is a natural number, then $n$ is an integer.

P: $n$ is a natural number

Q: $n$ is an integer

$\sim$P: $n$ is not a natural number

$\sim$Q: $n$ is not an integer

Contrapositive: If $n$ is not an integer, then $n$ is not a natural number.

(iii) If all three sides of a triangle are equal, then the triangle is equilateral.

P: All three sides of a triangle are equal

Q: The triangle is equilateral

$\sim$P: Not all three sides of a triangle are equal (or, At least one pair of sides are not equal)

$\sim$Q: The triangle is not equilateral

Contrapositive: If the triangle is not equilateral, then all three sides of the triangle are not equal.

(iv) If $x$ and $y$ are negative integers, then $xy$ is positive.

P: $x$ and $y$ are negative integers

Q: $xy$ is positive

$\sim$P: $x$ is not a negative integer or $y$ is not a negative integer

$\sim$Q: $xy$ is not positive (i.e., $xy \leq 0$)

Contrapositive: If $xy$ is not positive, then $x$ is not a negative integer or $y$ is not a negative integer.

(v) If natural number $n$ is divisible by 6, then $n$ is divisible by 2 and 3.

P: Natural number $n$ is divisible by 6

Q: $n$ is divisible by 2 and 3

$\sim$P: Natural number $n$ is not divisible by 6

$\sim$Q: $n$ is not divisible by 2 or $n$ is not divisible by 3

Contrapositive: If natural number $n$ is not divisible by 2 or $n$ is not divisible by 3, then natural number $n$ is not divisible by 6.

(vi) If it snows, then the weather will be cold.

P: It snows

Q: The weather will be cold

$\sim$P: It does not snow

$\sim$Q: The weather will not be cold

Contrapositive: If the weather will not be cold, then it does not snow.

(vii) If $x$ is a real number such that $0 < x < 1$, then $x^2 < 1$.

P: $x$ is a real number such that $0 < x < 1$

Q: $x^2 < 1$

$\sim$P: $x$ is not a real number such that $0 < x < 1$ (i.e., $x \leq 0$ or $x \geq 1$, assuming $x$ is a real number)

$\sim$Q: $x^2 \geq 1$

Contrapositive: If $x^2 \geq 1$, then $x$ is not a real number such that $0 < x < 1$. (Or, If $x^2 \geq 1$, then $x \leq 0$ or $x \geq 1$).

The converse of a conditional statement "If P, then Q" is formed by swapping the hypothesis (P) and the conclusion (Q). The converse is the statement "If Q, then P". Symbolically, the converse of $P \to Q$ is $Q \to P$.

(i) If a rectangle ‘R’ is a square, then R is a rhombus.

P: A rectangle R is a square.

Q: R is a rhombus.

Converse: If R is a rhombus, then a rectangle R is a square.

(ii) If today is Monday, then tomorrow is Tuesday.

P: Today is Monday.

Q: Tomorrow is Tuesday.

Converse: If tomorrow is Tuesday, then today is Monday.

(iii) If you go to Agra, then you must visit Taj Mahal.

P: You go to Agra.

Q: You must visit Taj Mahal.

Converse: If you must visit Taj Mahal, then you go to Agra.

(iv) If the sum of squares of two sides of a triangle is equal to the square of third side of a triangle, then the triangle is right angled.

P: The sum of squares of two sides of a triangle is equal to the square of third side of a triangle.

Q: The triangle is right angled.

Converse: If the triangle is right angled, then the sum of squares of two sides of a triangle is equal to the square of third side of a triangle.

(v) If all three angles of a triangle are equal, then the triangle is equilateral.

P: All three angles of a triangle are equal.

Q: The triangle is equilateral.

Converse: If the triangle is equilateral, then all three angles of a triangle are equal.

(vi) If $x : y = 3 : 2$, then $2x = 3y$.

P: $x : y = 3 : 2$

Q: $2x = 3y$

Converse: If $2x = 3y$, then $x : y = 3 : 2$.

(vii) If S is a cyclic quadrilateral, then the opposite angles of S are supplementary.

P: S is a cyclic quadrilateral.

Q: The opposite angles of S are supplementary.

Converse: If the opposite angles of S are supplementary, then S is a cyclic quadrilateral.

(viii) If x is zero, then x is neither positive nor negative.

P: x is zero.

Q: x is neither positive nor negative.

Converse: If x is neither positive nor negative, then x is zero.

(ix) If two triangles are similar, then the ratio of their corresponding sides are equal.

P: Two triangles are similar.

Q: The ratio of their corresponding sides are equal.

Converse: If the ratio of their corresponding sides are equal, then two triangles are similar.

Quantifiers are words or phrases that indicate the extent to which a predicate applies to a range of objects. The common quantifiers are the universal quantifier (e.g., "for all", "every") and the existential quantifier (e.g., "there exists", "some").

(i) There exists a triangle which is not equilateral.

The quantifier is "There exists".

(ii) For all real numbers x and y, xy = yx.

The quantifier is "For all".

(iii) There exists a real number which is not a rational number.

The quantifier is "There exists".

(iv) For every natural number x, x + 1 is also a natural number.

The quantifier is "For every".

(v) For all real numbers x with x > 3, x² is greater than 9.

The quantifier is "For all".

(vi) There exists a triangle which is not an isosceles triangle.

The quantifier is "There exists".

(vii) For all negative integers x, x³ is also a negative integers.

The quantifier is "For all".

(viii) There exists a statement in above statements which is not true.

The quantifier is "There exists".

(ix) There exists a even prime number other than 2.

The quantifier is "There exists".

(x) There exists a real number x such that $x^2 + 1 = 0$.

The quantifier is "There exists".

We want to prove that for any integer $n$, the expression $n^3 - n$ is always even using the direct method. An integer is even if it can be written in the form $2k$ for some integer $k$. An integer is odd if it can be written in the form $2k + 1$ for some integer $k$.

First, let's factor the expression $n^3 - n$:

$n^3 - n = n(n^2 - 1)$

Using the difference of squares formula ($a^2 - b^2 = (a-b)(a+b)$), we have $n^2 - 1 = (n-1)(n+1)$.

So, $n^3 - n = n(n-1)(n+1)$.

We can rewrite this as $(n-1)n(n+1)$. This is the product of three consecutive integers: $n-1$, $n$, and $n+1$.

Now, we consider the two cases for the integer $n$:

Case 1: $n$ is an even integer.

If $n$ is even, then $n$ can be written as $2k$ for some integer $k$.

Substituting this into the expression $(n-1)n(n+1)$, we get:

$n^3 - n = (2k - 1)(2k)(2k + 1)$

$n^3 - n = 2k(2k - 1)(2k + 1)$

Let $M = k(2k - 1)(2k + 1)$. Since $k$ is an integer, and the product and difference of integers are integers, $M$ is also an integer.

So, $n^3 - n = 2M$, where $M$ is an integer.

By the definition of an even number, this shows that if $n$ is even, then $n^3 - n$ is even.

Case 2: $n$ is an odd integer.

If $n$ is odd, then $n$ can be written as $2k + 1$ for some integer $k$.

Substituting this into the expression $(n-1)n(n+1)$, we get:

$n-1 = (2k + 1) - 1 = 2k$

$n+1 = (2k + 1) + 1 = 2k + 2 = 2(k+1)$

So, $n^3 - n = (n-1)n(n+1) = (2k)(2k+1)(2(k+1))$

$n^3 - n = 2k \cdot (2k+1) \cdot 2(k+1)$

$n^3 - n = 2 \cdot [k(2k+1) \cdot 2(k+1)]$

$n^3 - n = 2 \cdot [2k(k+1)(2k+1)]$

Let $N = 2k(k+1)(2k+1)$. Since $k$ is an integer, the product and sum of integers are integers, so $N$ is an integer.

So, $n^3 - n = 2N$, where $N$ is an integer.

By the definition of an even number, this shows that if $n$ is odd, then $n^3 - n$ is even.

In both cases (when $n$ is even and when $n$ is odd), we have shown that $n^3 - n$ is an even number.

Therefore, for any integer $n$, $n^3 - n$ is always even.

To check the validity of a compound statement, we determine the truth values of its component statements and then use the rules of logical connectives.

(i) p : 125 is divisible by 5 and 7.

This is a compound statement with the connective "and". Let's break it down into component statements:

Component 1 (A): 125 is divisible by 5.

Component 2 (B): 125 is divisible by 7.

Let's determine the truth value of each component:

For Component 1 (A): $125 \div 5 = 25$. The remainder is 0. So, 125 is divisible by 5. Statement A is True.

For Component 2 (B): $125 = 7 \times 17 + 6$. The remainder is 6, not 0. So, 125 is not divisible by 7. Statement B is False.

The compound statement is of the form "A and B". An "and" statement is true only if both component statements are true. Since Statement B is false, the compound statement "125 is divisible by 5 and 7" is False.

The validity of the statement is False (or Invalid).

(ii) q : 131 is a multiple of 3 or 11.

This is a compound statement with the connective "or". Let's break it down into component statements:

Component 1 (A): 131 is a multiple of 3.

Component 2 (B): 131 is a multiple of 11.

Let's determine the truth value of each component:

For Component 1 (A): To check if 131 is a multiple of 3, we can sum its digits: $1+3+1 = 5$. Since 5 is not divisible by 3, 131 is not divisible by 3. Statement A is False.

For Component 2 (B): Let's divide 131 by 11. $131 = 11 \times 11 + 10$. The remainder is 10, not 0. So, 131 is not a multiple of 11. Statement B is False.

The compound statement is of the form "A or B". An "or" statement is true if at least one component statement is true. It is false only if both component statements are false. Since both Statement A and Statement B are false, the compound statement "131 is a multiple of 3 or 11" is False.

The validity of the statement is False (or Invalid).

We want to prove the statement "The sum of an irrational number and a rational number is irrational" using the method of contradiction.

Let the given statement be $p$: The sum of an irrational number and a rational number is irrational.

Assume the negation of the statement $p$, denoted by $\sim p$, is true.

$\sim p$: The sum of an irrational number and a rational number is rational.

Let $I$ be an irrational number and $R$ be a rational number.

According to our assumption ($\sim p$), the sum $I + R$ is a rational number. Let's call this sum $S$. So, we have:

$I + R = S$

By the definition of a rational number, a number is rational if it can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

Since $R$ is a rational number, we can write $R = \frac{a}{b}$ for some integers $a$ and $b$, where $b \neq 0$.

Since, by our assumption, $S$ is a rational number, we can write $S = \frac{c}{d}$ for some integers $c$ and $d$, where $d \neq 0$.

Now, substitute the expressions for $R$ and $S$ into the equation $I + R = S$:

$I + \frac{a}{b} = \frac{c}{d}$

We want to express $I$ in terms of rational numbers. Subtract $\frac{a}{b}$ from both sides of the equation:

$I = \frac{c}{d} - \frac{a}{b}$

To subtract the fractions on the right side, we find a common denominator, which is $bd$:

$I = \frac{c \cdot b}{d \cdot b} - \frac{a \cdot d}{b \cdot d}$

$I = \frac{cb - ad}{bd}$

Since $a, b, c,$ and $d$ are integers, the numerator $cb - ad$ is also an integer (the difference and product of integers are integers).

Since $b \neq 0$ and $d \neq 0$, their product $bd$ is also a non-zero integer (the product of two non-zero numbers is non-zero).

Thus, the expression for $I$ is a ratio of two integers with a non-zero denominator.

$I = \frac{\text{integer}}{\text{non-zero integer}}$

By the definition of a rational number, this means that $I$ is a rational number.

This contradicts our initial premise that $I$ is an irrational number.

Since our assumption that the sum of an irrational number and a rational number is rational leads to a contradiction, the assumption must be false.

Therefore, the original statement $p$ must be true.

Thus, the sum of an irrational number and a rational number is irrational.

We are asked to prove the statement: For any real numbers $x, y$, if $x = y$, then $x^2 = y^2$.

We will use the direct method of proof.

Assume that $x$ and $y$ are real numbers such that $x = y$. This is our hypothesis.

We want to show that the conclusion, $x^2 = y^2$, follows directly from this assumption.

Consider the left-hand side of the equation we want to prove, $x^2$. By definition of squaring, $x^2 = x \cdot x$.

Since we are given that $x = y$, by the property of substitution in equality, we can replace any occurrence of $x$ with $y$. Let's replace the second factor of $x$ in the expression $x \cdot x$ with $y$ (since $x=y$).

So, $x^2 = x \cdot (y)$.

Now, again using the fact that $x = y$, we can replace the first factor of $x$ in the expression $x \cdot y$ with $y$ (since $x=y$).

So, $x^2 = (y) \cdot y$.

By the definition of squaring, $y \cdot y = y^2$.

Therefore, we have $x^2 = y^2$.

Thus, starting with the assumption that $x = y$, we have logically derived that $x^2 = y^2$ using algebraic properties of real numbers and the principle of substitution.

Therefore, by the direct method, for any real numbers $x, y$, if $x = y$, then $x^2 = y^2$.

We want to prove the statement "If $n^2$ is an even integer, then $n$ is also an even integer" using the contrapositive method.

The original statement is in the form "If P, then Q", where:

• P: $n^2$ is an even integer.
• Q: $n$ is an even integer.

The contrapositive of the statement "If P, then Q" is "If not Q, then not P".

• Not Q ($\sim Q$): $n$ is not an even integer. The negation of "n is an even integer" is "n is an odd integer".
• Not P ($\sim P$): $n^2$ is not an even integer. The negation of "n$^2$ is an even integer" is "$n^2$ is an odd integer".

So, the contrapositive statement is: If $n$ is an odd integer, then $n^2$ is an odd integer.

Now, we will prove this contrapositive statement using the direct method.

Assume the hypothesis of the contrapositive is true:

Let $n$ be an odd integer.

By the definition of an odd integer, $n$ can be written in the form $2k + 1$ for some integer $k$.

So, we have $n = 2k + 1$, where $k \in \mathbb{Z}$.

Now, we need to show that $n^2$ is odd. Let's calculate $n^2$ by substituting the expression for $n$:

$n^2 = (2k + 1)^2$

Expand the right-hand side:

$n^2 = (2k)^2 + 2(2k)(1) + 1^2$

$n^2 = 4k^2 + 4k + 1$

To show that $n^2$ is odd, we need to express it in the form $2m + 1$ for some integer $m$. We can factor out 2 from the first two terms:

$n^2 = 2(2k^2 + 2k) + 1$

Let $m = 2k^2 + 2k$. Since $k$ is an integer, $k^2$ is an integer, $2k^2$ is an integer, and $2k$ is an integer. The sum of two integers is also an integer. Therefore, $m = 2k^2 + 2k$ is an integer.

So, we have $n^2 = 2m + 1$, where $m$ is an integer.

By the definition of an odd integer, this shows that $n^2$ is an odd integer.

We have successfully proven the contrapositive statement: If $n$ is an odd integer, then $n^2$ is an odd integer.

Since the contrapositive statement is logically equivalent to the original statement, the original statement is also true.

Conclusion: By the contrapositive method, we have proven that if $n^2$ is an even integer, then $n$ is also an even integer.

Answer: (C)

Explanation:

A statement in logic is a declarative sentence that is either true or false, but not both.

(A) "x is a real number." is an open sentence, whose truth value depends on the value of the variable x.

(B) "Switch off the fan." is an imperative sentence (a command).

(D) "Let me go." is an imperative sentence (a request).

(C) "6 is a natural number." is a declarative sentence that is true. Therefore, it is a statement.

Answer: (D)

Explanation:

A statement is a declarative sentence that has a definite truth value (either true or false).

(A) "Smoking is injurious to health." is a declarative sentence which is generally considered true. It is a statement.

(B) "2 + 2 = 4" is a declarative sentence which is true. It is a statement.

(C) "2 is the only even prime number." is a declarative sentence which is true. It is a statement.

(D) "Come here." is an imperative sentence (a command). It is not a declarative sentence and does not have a truth value. Therefore, it is not a statement.

Explanation:

The given statement is "2 + 7 > 9 or 2 + 7 < 9".

This statement connects two simpler mathematical inequalities using the word "or".

The connectives in logic are words like "and", "or", "if...then", "if and only if", and "not".

In this statement, the word "or" serve

Answer: (D)

Explanation:

The given statement is "Earth revolves round the Sun and Moon is a satellite of earth".

This statement is a compound statement formed by joining two simple statements:

Statement 1: "Earth revolves round the Sun"

Statement 2: "Moon is a satellite of earth"

These two statements are connected by the word "and".

The word "and" is a logical connective used to combine two or more statements.

Answer: (C)

Explanation:

The negation of a statement simply asserts the opposite of the original statement.

If the statement is "P", its negation is "not P".

Given statement: "A circle is an ellipse".

The negation of this statement is "A circle is not an ellipse".

Answer: (B)

Explanation:

The given statement is "7 is greater than 8". This can be written as $7 > 8$.

The negation of a statement asserts the opposite.

The negation of "$7 > 8$" is "it is not the case that $7 > 8$".

This is equivalent to "$7 \not> 8$".

The inequality "$7 \not> 8$" means that 7 is not greater than 8, which includes the possibilities that 7 is less than 8 ($7 < 8$) or 7 is equal to 8 ($7 = 8$). So, "$7 \not> 8$" is the same as "$7 \leq 8$".

Looking at the options:

(A) "$7 = 8$" is a specific case that is implied by the negation, but it is not the complete negation itself.

(B) "7 is not greater than 8" is the direct negation of the statement "7 is greater than 8".

(C) "8 is less than 7" is the same as "7 is greater than 8", which is the original statement.

Therefore, the correct negation is "7 is not greater than 8".

Answer: (A)

Explanation:

Let $P$ be the statement "72 is divisible by 2".

Let $Q$ be the statement "72 is divisible by 3".

The given statement is of the form "$P \text{ and } Q$".

The negation of a compound statement of the form "$P \text{ and } Q$" is given by De Morgan's Law as "$\text{not } P \text{ or not } Q$".

The negation of $P$ is "72 is not divisible by 2".

The negation of $Q$ is "72 is not divisible by 3".

Therefore, the negation of the statement "72 is divisible by 2 and 3" is "72 is not divisible by 2 or 72 is not divisible by 3".

This matches option (A).

Answer: (B)

Explanation:

Let $P$ be the statement "Plants take in CO₂".

Let $Q$ be the statement "Plants give out O₂".

The given statement is of the form "$P \text{ and } Q$".

According to De Morgan's Law, the negation of "$P \text{ and } Q$" is "$\text{not } P \text{ or not } Q$".

The negation of $P$ is "Plants do not take in CO₂".

The negation of $Q$ is "Plants do not give out O₂".

Combining these with "or", the negation is "Plants do not take in CO₂ or plants do not give out O_2}$".

Answer: (C)

Explanation:

Let $P$ be the statement "Rajesh lived in Bangalore".

Let $Q$ be the statement "Rajni lived in Bangalore".

The given statement is of the form "$P \text{ or } Q$".

According to De Morgan's Law, the negation of "$P \text{ or } Q$" is "$\text{not } P \text{ and not } Q$".

The negation of $P$ is "Rajesh did not live in Bangalore".

The negation of $Q$ is "Rajni did not live in Bangalore".

Combining these with "and", the negation is "Rajesh did not live in Bangalore and Rajni did not live in Bangalore".

This matches option (C).

Answer: (A)

Explanation:

The negation of a statement simply asserts the opposite of the original statement.

If the statement is "P", its negation is "not P".

If the statement is "not P", its negation is "P".

The given statement is "101 is not a multiple of 3". This is a statement in the form "not P", where P is the statement "101 is a multiple of 3".

The negation of "not P" is P.

Therefore, the negation of "101 is not a multiple of 3" is "101 is a multiple of 3".

Answer: (C)

Explanation:

A conditional statement is of the form "If P, then Q".

The contrapositive of the statement "If P, then Q" is "If not Q, then not P".

In the given statement: "If 7 is greater than 5, then 8 is greater than 6"

Let P be the statement: "7 is greater than 5"

Let Q be the statement: "8 is greater than 6"

The negation of P (not P) is: "7 is not greater than 5"

The negation of Q (not Q) is: "8 is not greater than 6"

The contrapositive is "If not Q, then not P".

Substituting the negations, we get: "If 8 is not greater than 6, then 7 is not greater than 5".

Answer: (B)

Explanation:

A conditional statement is of the form "If P, then Q".

The converse of the statement "If P, then Q" is "If Q, then P".

In the given statement: "If $x > y$, then $x + a > y + a$"

Let P be the statement: "$x > y$"

Let Q be the statement: "$x + a > y + a$"

The converse is "If Q, then P".

Substituting the statements, we get: "If $x + a > y + a$, then $x > y$".

Answer: (A)

Explanation:

A conditional statement is of the form "If P, then Q".

The converse of the statement "If P, then Q" is "If Q, then P".

In the given statement: “If sun is not shining, then sky is filled with clouds”

Let P be the statement: "sun is not shining"

Let Q be the statement: "sky is filled with clouds"

The converse is "If Q, then P".

Substituting the statements, we get: "If sky is filled with clouds, then sun is not shining".

This matches option (A).

Option (C) is the contrapositive of the original statement.

Answer: (C)

Explanation:

A conditional statement is of the form "If P, then Q".

The contrapositive of the statement "If P, then Q" is "If not Q, then not P".

In the given statement: "If p, then q"

Let P be the statement "p".

Let Q be the statement "q".

The negation of P is "~p".

The negation of Q is "~q".

The contrapositive is "If not Q, then not P".

Substituting the negations, we get: "If ~q, then ~p".

This matches option (C).

Answer: (B)

Explanation:

Let the given statement be S: "If $x^2$ is not even, then $x$ is not even".

This statement is in the form "If Q, then P", where:

Q: "$x^2$ is not even"

P: "$x$ is not even"

We are given that statement S is the converse of some original statement.

The converse of a statement "If A, then B" is "If B, then A".

If S is the converse of an original statement "If A, then B", then S must be "If B, then A".

Comparing S ("If Q, then P") with the form of a converse ("If B, then A"), we can see that B corresponds to Q and A corresponds to P.

So, the original statement is "If A, then B", which is "If P, then Q".

Substituting the statements for P and Q:

P is "$x$ is not even".

Q is "$x^2$ is not even".

The original statement is "If $x$ is not even, then $x^2$ is not even".

Now, let's examine the options:

(A) "If $x^2$ is odd, then $x$ is even." (Note that "$x^2$ is not even" is equivalent to "$x^2$ is odd", and "$x$ is not even" is equivalent to "$x$ is odd". So this option is "If $x^2$ is not even, then $x$ is not not even"). This is not the original statement.

(B) "If $x$ is not even, then $x^2$ is not even." This matches our derived original statement.

(C) "If $x$ is even, then $x^2$ is even." (This is the contrapositive of the original statement, and also equivalent to "If not P, then not Q").

(D) "If $x$ is odd, then $x^2$ is even." (This is "If P, then not Q").

Therefore, the statement "If $x^2$ is not even, then $x$ is not even" is the converse of the statement "If $x$ is not even, then $x^2$ is not even".

Answer: (A)

Explanation:

A conditional statement is of the form "If P, then Q".

The contrapositive of the statement "If P, then Q" is "If not Q, then not P".

In the given statement: “If Chandigarh is capital of Punjab, then Chandigarh is in India”

Let P be the statement: "Chandigarh is capital of Punjab"

Let Q be the statement: "Chandigarh is in India"

The negation of P (not P) is: "Chandigarh is not capital of Punjab"

The negation of Q (not Q) is: "Chandigarh is not in India"

The contrapositive is "If not Q, then not P".

Substituting the negations, we get: "If Chandigarh is not in India, then Chandigarh is not the capital of Punjab".

Answer: (C)

Explanation:

The conditional statement $p \to q$ means "If p, then q".

Let's analyze each option:

(A) "q is sufficient for p" means that the truth of q is enough to guarantee the truth of p. This is represented as $q \to p$.

(B) "p is necessary for q" means that q can only be true if p is true. This is represented as $q \to p$. An equivalent phrasing is "If not p, then not q", which is the contrapositive of $q \to p$.

(C) "p only if q" means that p can only be true if q is true. If p is true, then q must be true. This is represented as $p \to q$. An equivalent phrasing is "If not q, then not p", which is the contrapositive of $p \to q$.

(D) "if q, then p" is directly represented as $q \to p$.

Therefore, the phrase "p only if q" represents the conditional statement $p \to q$.

Answer: (A)

Explanation:

Let the given statement be P: "The product of 3 and 4 is 9".

The negation of a statement P is "It is false that P" or "not P".

The negation of "The product of 3 and 4 is 9" can be stated as "It is false that the product of 3 and 4 is 9".

Alternatively, it can be stated as "The product of 3 and 4 is not 9".

Let's check the options:

(A) "It is false that the product of 3 and 4 is 9." This is a direct form of negation.

(B) "The product of 3 and 4 is 12." While true, this statement is not the logical negation of the original statement. The negation must directly contradict the original claim about the product being 9.

(C) "The product of 3 and 4 is not 12." This is the negation of the statement "The product of 3 and 4 is 12".

(D) "It is false that the product of 3 and 4 is not 9." This is the negation of the statement "The product of 3 and 4 is not 9", which is equivalent to the original statement itself.

Thus, option (A) correctly represents the negation of the given statement.

Answer: (C)

Explanation:

Let the original statement be P: "A natural number is greater than zero".

The negation of statement P (denoted as $\sim$P) is "A natural number is not greater than zero".

Let's evaluate the options:

(A) "A natural number is not greater than zero." This is the direct negation of P ($\sim$P).

(B) "It is false that a natural number is greater than zero." This is equivalent to saying "not (A natural number is greater than zero)", which is $\sim$P.

(C) "It is false that a natural number is not greater than zero." Let Q be the statement "A natural number is not greater than zero". This option is "It is false that Q", which is $\sim$Q. Since Q is $\sim$P, $\sim$Q is $\sim(\sim$P), which is equivalent to P. So, option (C) is equivalent to the original statement P, not its negation.

(D) "None of the above." Since option (C) is not a negation of the original statement, this option is incorrect.

Therefore, option (C) is the one that is not a negation of the original statement.

Answer: (B)

Explanation:

A conjunction is a compound statement formed by joining two or more simple statements using the logical connective "and". The resulting conjunction is true if and only if all the simple statements are true.

Let's analyze the options:

(A) "Ram and Shyam are friends." This statement describes a relationship between Ram and Shyam. While it uses "and", it is often treated as a single relational proposition rather than a conjunction of two distinct properties ("Ram is a friend" and "Shyam is a friend" - as the latter statements are incomplete without specifying 'friend of whom').

(B) "Both Ram and Shyam are tall." This statement can be logically decomposed into two simple statements joined by "and": "Ram is tall" and "Shyam is tall". If we let P be "Ram is tall" and Q be "Shyam is tall", the statement is equivalent to "P and Q" ($P \land Q$). This is a conjunction.

(C) "Both Ram and Shyam are enemies." Similar to option (B), this statement is equivalent to "Ram is an enemy and Shyam is an enemy". This is also a conjunction.

Since both (B) and (C) fit the definition of a conjunction, and assuming this is a single-choice question with only one correct answer, there might be an issue with the question or the options provided. However, option (B) is a clear and common example of a sentence that translates into a logical conjunction. If only one is to be selected, either (B) or (C) would be valid. Let's assume (B) is the intended answer.

A sentence is called a mathematical statement if it is either true or false, but not both.

(i) The angles opposite to equal sides of a triangle are equal.
This sentence is always true.
Hence, it is a statement.

(ii) The moon is a satellite of earth.
This sentence is always true.
Hence, it is a statement.

(iii) May God bless you!
This sentence is an optative sentence (expressing a wish). It cannot be judged as true or false.
Hence, it is not a statement.

(iv) Asia is a continent.
This sentence is always true.
Hence, it is a statement.

(v) How are you?
This sentence is an interrogative sentence (a question). It cannot be judged as true or false.
Hence, it is not a statement.