Multiple Correct Answers MCQs for Sub-Topics of Topic 11: Mathematical Reasoning

Question 1. Which of the following are mathematical statements (propositions)?

(A) $5 + 7 = 13$

(B) Look out!

(C) The Earth revolves around the Sun.

(D) Is $x > 5$?

(E) Every square is a rhombus.

(F) What is the price of this saree in Rupees?

Question 2. Identify the sentences that are NOT mathematical statements (propositions).

(A) $\sqrt{3}$ is an irrational number.

(B) Please bring me a glass of water.

(C) If $x > 0$, then $x^2 > 0$.

(D) How beautiful the sunset is!

(E) $x + y = 10$

(F) Close your eyes.

Question 3. Which of the following sentences can be assigned a truth value (True or False)?

(A) Every prime number greater than $2$ is odd.

(B) Will you come to the party?

(C) The population of India is over $1.4$ billion.

(D) This statement is false.

(E) $7 - 2 = 6$

Question 4. Which of the following are open sentences (not propositions) as they stand?

(A) $x^2 + 3x + 2 = 0$

(B) He is a famous cricketer.

(C) The square root of $9$ is $3$.

(D) It is cold outside.

(E) $y < 0$

Question 5. A sentence is a proposition if it:

(A) Is grammatically correct.

(B) Is declarative.

(C) Has a definite truth value.

(D) Contains mathematical symbols.

(E) Expresses a command.

Question 6. Which of the following statements are true?

(A) The sum of any two even numbers is even.

(B) The square of every integer is positive.

(C) Every triangle is a right-angled triangle.

(D) $100$ is a multiple of $5$.

(E) $\pi$ is a rational number.

Question 7. Which of the following statements are false?

(A) February has $30$ days.

(B) The smallest prime number is $1$.

(C) The sum of angles in a quadrilateral is $180^\circ$.

(D) All radii of a circle have the same length.

(E) $-5$ is an natural number.

Question 8. A statement is considered false if:

(A) Its negation is true.

(B) It leads to a contradiction.

(C) It is not a question.

(D) It cannot be proven true.

(E) It asserts something that is not the case in reality or mathematics.

Question 9. Consider the sentence: "This sentence is a question." Is this a proposition? Explain.

(A) Yes, its truth value is True.

(B) Yes, its truth value is False.

(C) No, because it refers to itself.

(D) No, because it is a question.

(E) No, because it leads to a paradox.

Question 10. Which of the following sentences have a definite truth value?

(A) The atomic number of Hydrogen is $1$.

(B) $2x + 1 = 5$

(C) The capital of Tamil Nadu is Chennai.

(D) Sit down.

(E) The largest city in Rajasthan is Jaipur.

Question 11. What defines a mathematical statement (proposition)?

(A) It must contain numbers or symbols.

(B) It must be a declarative sentence.

(C) It must be objectively either true or false.

(D) It must be a universal truth.

(E) It must be provable.

Question 12. Which of the following is a true statement?

(A) All triangles are equilateral.

(B) Some prime numbers are even.

(C) The set of integers is a subset of the set of rational numbers.

(D) The sum of two negative integers is always positive.

(E) Every rhombus is a parallelogram.

Question 13. Which of the following sentences cannot be classified as either true or false without additional information?

(A) He lives in Mumbai.

(B) $x > 10$

(C) The chemical formula for water is $\text{H}_2\text{O}$.

(D) She is a good singer.

(E) The current Prime Minister of India is Narendra Modi.

Question 14. A statement is sometimes called a:

(A) Argument

(B) Predicate

(C) Proposition

(D) Assertion

(E) Sentence

Question 15. Which of the following descriptions apply to a valid mathematical statement?

(A) It is an expression of emotion.

(B) It is a declarative sentence.

(C) Its truth value is objective.

(D) It must be mathematically relevant.

(E) Its truth value is subjective.

Question 1. Which of the following are correct negations of the statement "All birds can fly"?

(A) Some birds cannot fly.

(B) No birds can fly.

(C) Not all birds can fly.

(D) There exists a bird that cannot fly.

(E) All birds cannot fly.

Question 2. If $p$ is the statement "The weather is hot", which of the following represent $\neg p$?

(A) The weather is not hot.

(B) It is false that the weather is hot.

(C) The weather is cold.

(D) It is not true that the weather is hot.

(E) The weather is cool.

Question 3. Which of the following are compound statements?

(A) The cat is black.

(B) The cat is black and the dog is brown.

(C) If it is raining, then I will stay indoors.

(D) The number $7$ is prime or even.

(E) Pass the book.

Question 4. If a statement $p$ is true, then its negation $\neg p$ must be:

(A) True

(B) False

(C) Undetermined

(D) A proposition

(E) A simple statement

Question 5. The negation of "Some students like Maths" is:

(A) Some students do not like Maths.

(B) All students like Maths.

(C) No student likes Maths.

(D) It is false that some students like Maths.

(E) Every student does not like Maths.

Question 6. Which logical connectives are used to form compound statements from simple statements?

(A) $\land$ (AND)

(B) $\neg$ (NOT)

(C) $\lor$ (OR)

(D) $\implies$ (IF...THEN)

(E) $\iff$ (IF AND ONLY IF)

Question 7. The negation of the statement "It is cold and it is raining" is logically equivalent to:

(A) It is not cold and it is not raining.

(B) It is not cold or it is not raining.

(C) It is cold or it is raining.

(D) If it is cold, then it is not raining.

(E) Neither it is cold nor it is raining.

Question 8. If a statement $p$ is false, then which of the following statements must be true?

(A) $p$

(B) $\neg p$

(C) $\neg (\neg p)$

(D) $p \land q$ (for any $q$)

(E) $p \lor q$ (for any $q$ that is true)

Question 9. Which of the following correctly describes the negation of a statement?

(A) It has the opposite meaning of the original statement.

(B) It reverses the order of words in the statement.

(C) It has the opposite truth value of the original statement.

(D) It is formed by adding "not" to the original statement.

(E) It contradicts the original statement.

Question 10. Let $p$ be "The number is even" and $q$ be "The number is a multiple of $2$". Which of the following are compound statements formed from $p$ and $q$?

(A) The number is even and a multiple of $2$.

(B) The number is even or a multiple of $2$.

(C) The number is not even.

(D) If the number is even, then it is a multiple of $2$.

(E) Is the number even?

Question 11. The symbol $\neg$ is used for:

(A) Conjunction

(B) Disjunction

(C) Negation

(D) Denial

(E) Complement

Question 12. The negation of the statement "There is a student who scored $100\%$ marks" is:

(A) There is a student who did not score $100\%$ marks.

(B) No student scored $100\%$ marks.

(C) All students scored $100\%$ marks.

(D) Every student did not score $100\%$ marks.

(E) Not every student scored $100\%$ marks.

Question 13. If $p$ is true and $q$ is false, which of the following compound statements are false?

(A) $p \land q$

(B) $p \lor q$

(C) $\neg p \land q$

(D) $\neg p \lor q$

(E) $p \land \neg q$

Question 14. Which of the following are examples of negating a simple statement?

(A) Turning "It is hot" into "It is not hot".

(B) Turning "$2+2=4$" into "$2+2 \neq 4$".

(C) Turning "The grass is green" into "The grass is brown".

(D) Turning "The door is open" into "The door is closed".

(E) Turning "All cats are black" into "Some cats are not black".

Question 15. Which of the following statements about negation are true?

(A) The negation of a true statement is false.

(B) The negation of a false statement is true.

(C) $\neg (\neg p)$ is logically equivalent to $p$.

(D) The negation symbol $\neg$ connects two statements.

(E) Negation is a unary operation.

Question 1. Which of the following statements about the logical connective $\land$ (AND) are true?

(A) $p \land q$ is true if and only if both $p$ and $q$ are true.

(B) $p \land q$ is false if at least one of $p$ or $q$ is false.

(C) $\land$ is also known as conjunction.

(D) The truth table column for $p \land q$ when $p$ is T, F, T, F and $q$ is T, T, F, F is T, F, F, F.

(E) $p \land q$ is logically equivalent to $q \land p$.

Question 2. Which of the following statements about the logical connective $\lor$ (OR) are true (using inclusive OR)?

(A) $p \lor q$ is true if at least one of $p$ or $q$ is true.

(B) $p \lor q$ is false if and only if both $p$ and $q$ are false.

(C) $\lor$ is also known as disjunction.

(D) The truth table column for $p \lor q$ when $p$ is T, F, T, F and $q$ is T, T, F, F is T, T, T, F.

(E) $p \lor q$ is logically equivalent to $q \lor p$.

Question 3. Given $p$ is true, $q$ is false, and $r$ is true. What is the truth value of the following compound statements?

(A) $p \land q$ is False.

(B) $p \lor q$ is True.

(C) $\neg r$ is False.

(D) $(\neg p) \land r$ is True.

(E) $p \lor (\neg q)$ is False.

Question 4. Which of the following truth table columns correctly represents the result of applying a standard logical connective to $p$ and $q$? (Assume $p$ columns are T, T, F, F and $q$ columns are T, F, T, F)

(A) $p \land q$: T, F, F, F

(B) $p \lor q$: F, F, F, T

(C) $\neg p$: F, F, T, T

(D) $\neg q$: F, T, F, T

(E) $p \land \neg q$: F, T, F, F

Question 5. If a compound statement involves $n$ distinct simple propositions, how many rows are needed in its truth table?

(A) $n$ rows

(B) $n^2$ rows

(C) $2^n$ rows

(D) A number that depends on the connectives used.

(E) $2 \times n$ rows

Question 6. The exclusive OR (XOR), denoted by $p \oplus q$, is logically equivalent to which of the following?

(A) $\neg (p \iff q)$

(B) $(p \lor q) \land \neg (p \land q)$

(C) $(p \land \neg q) \lor (\neg p \land q)$

(D) $p \lor q$ where $p$ and $q$ are mutually exclusive.

(E) $(p \land q) \lor (\neg p \land \neg q)$

Question 7. Given the truth values for $p$ and $q$ below, which statements about $p \land q$ and $p \lor q$ (inclusive) are correct?

p | q | p $\land$ q | p $\lor$ q

T | T | T | T

T | F | F | T

F | T | F | T

F | F | F | F

(A) When $p$ and $q$ are both false, $p \land q$ is false.

(B) When $p$ and $q$ are both true, $p \lor q$ is true.

(C) $p \land q$ is true in more cases than $p \lor q$.

(D) $p \lor q$ is false only when $p$ and $q$ are both false.

(E) $p \land q$ is true only when $p$ and $q$ are both true.

Question 8. Which of the following describes how to construct a truth table for a compound statement involving three simple propositions $p, q, r$?

(A) List all $2^3=8$ possible combinations of truth values for $p, q, r$.

(B) Add columns for intermediate compound statements (e.g., $\neg p$, $p \land q$).

(C) Evaluate the truth value of the main compound statement for each row.

(D) The final column will show the truth value of the compound statement for all possible inputs.

(E) Only the rows where the compound statement is true are necessary.

Question 9. If $p$ represents "I will buy a car" and $q$ represents "I will buy a bike", which of the following represent the statement "I will buy a car or a bike"? (Assuming inclusive OR).

(A) $p \lor q$

(B) I will buy a car OR I will buy a bike.

(C) $p \land q$

(D) It is not the case that I will not buy a car AND I will not buy a bike.

(E) $\neg (\neg p \land \neg q)$

Question 10. Consider the statement "The number is an integer greater than $0$ and less than $10$". Let $P(x)$ be "$x$ is an integer", $Q(x)$ be "$x > 0$", and $R(x)$ be "$x < 10$". Which of the following correctly symbolise parts or the whole statement?

(A) $P(x) \land Q(x) \land R(x)$

(B) $x > 0 \land x < 10$ (assuming $x$ is an integer)

(C) $\neg (x \leq 0 \lor x \geq 10)$ (assuming $x$ is an integer)

(D) $x \in \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$

(E) $x > 0 \lor x < 10$

Question 11. Which of the following are properties of the conjunction ($\land$) and disjunction ($\lor$) connectives?

(A) Commutative laws: $p \land q \equiv q \land p$ and $p \lor q \equiv q \lor p$

(B) Associative laws: $(p \land q) \land r \equiv p \land (q \land r)$ and $(p \lor q) \lor r \equiv p \lor (q \lor r)$

(C) Distributive laws: $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$ and $p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$

(D) Identity laws: $p \land \text{True} \equiv \text{True}$ and $p \lor \text{False} \equiv \text{False}$

(E) Idempotent laws: $p \land p \equiv p$ and $p \lor p \equiv p$

Question 12. How many distinct truth value combinations are there for a compound statement involving $p, q, r, s$?

(A) 4

(B) 8

(C) 16

(D) $2^4$

(E) Depends on the specific statement.

Question 13. Which of the following statements about exclusive OR ($p \oplus q$) are true?

(A) It is true when $p$ and $q$ have the same truth value.

(B) It is true when $p$ and $q$ have different truth values.

(C) It is false when $p$ and $q$ are both true.

(D) It is false when $p$ and $q$ are both false.

(E) It is the same as inclusive OR when one of the statements is true and the other is false.

Question 14. Consider the expression $(p \land \neg q) \lor (\neg p \land q)$. What is its truth value when $p$ is true and $q$ is false?

(A) True

(B) False

(C) Same as $p \land q$

(D) Same as $p \lor q$

(E) Same as $p \oplus q$

Question 15. Which of the following connectives are binary connectives (connecting two statements)?

(A) Negation ($\neg$)

(B) Conjunction ($\land$)

(C) Disjunction ($\lor$)

(D) Conditional ($\implies$)

(E) Biconditional ($\iff$)

Question 1. Which of the following sentences can be correctly represented by the form $p \implies q$?

(A) If you work hard, you will succeed.

(B) Working hard implies success.

(C) Success is guaranteed if you work hard.

(D) You will succeed only if you work hard.

(E) You work hard and you succeed.

Question 2. The conditional statement $p \implies q$ is false if and only if:

(A) $p$ is true and $q$ is true.

(B) $p$ is false and $q$ is false.

(C) $p$ is true and $q$ is false.

(D) The antecedent is true and the consequent is false.

(E) $p$ is a sufficient condition for $q$ is false.

Question 3. Which of the following are equivalent ways to state "$p \iff q$"?

(A) $p$ if and only if $q$.

(B) If $p$ then $q$, and if $q$ then $p$.

(C) $p$ is a necessary and sufficient condition for $q$.

(D) $p$ and $q$ have the same truth value.

(E) $\neg p$ if and only if $\neg q$.

Question 4. Consider the statement "If a quadrilateral is a square, then it is a rectangle". Let $p$ be "a quadrilateral is a square" and $q$ be "it is a rectangle". Which of the following are true?

(A) The statement $p \implies q$ is true.

(B) The statement $q \implies p$ (converse) is true.

(C) The statement $\neg q \implies \neg p$ (contrapositive) is true.

(D) The statement $\neg p \implies \neg q$ (inverse) is true.

(E) The statement $p \iff q$ (biconditional) is true.

Question 5. The biconditional statement $p \iff q$ is true when:

(A) $p$ is true and $q$ is true.

(B) $p$ is false and $q$ is false.

(C) $p$ and $q$ have different truth values.

(D) The truth table column for $p \iff q$ is T, F, F, T (for p: TTFF, q: TFTF).

(E) $p$ is logically equivalent to $q$.

Question 6. In the conditional statement "If you live in Bengaluru, then you live in Karnataka", which of the following are correct?

(A) "You live in Bengaluru" is the antecedent.

(B) "You live in Karnataka" is the consequent.

(C) This statement is false.

(D) This statement is true.

(E) The converse is "If you live in Karnataka, then you live in Bengaluru".

Question 7. The statement "$q$ is a necessary condition for $p$" is logically equivalent to which of the following?

(A) $p \implies q$

(B) $q \implies p$

(C) If not $q$, then not $p$.

(D) $\neg q \implies \neg p$

(E) $p$ only if $q$.

Question 8. The statement "$p$ is a sufficient condition for $q$" is logically equivalent to which of the following?

(A) $p \implies q$

(B) $q \implies p$

(C) If $p$, then $q$.

(D) $p$ implies $q$.

(E) $q$ whenever $p$.

Question 9. If the conditional statement "$p \implies q$" is true, which of the following must also be true?

(A) The converse ($q \implies p$)

(B) The inverse ($\neg p \implies \neg q$)

(C) The contrapositive ($\neg q \implies \neg p$)

(D) The negation ($p \land \neg q$)

(E) $\neg p \lor q$

Question 10. Which of the following statements about conditional and biconditional statements are correct?

(A) $p \iff q$ is true only when $p \implies q$ is true.

(B) $p \implies q$ is false only when $p$ is true and $q$ is false.

(C) $p \iff q$ is logically equivalent to $(p \implies q) \land (q \implies p)$.

(D) If $p \implies q$ is true, and $p$ is true, then $q$ must be true.

(E) If $p \iff q$ is true, then $p$ and $q$ have the same truth value.

Question 11. Let $p$ be "It is Sunday" and $q$ be "The shops are closed". Which of the following correctly represent the statement "The shops are closed if it is Sunday"?

(A) $p \implies q$

(B) If it is Sunday, then the shops are closed.

(C) $q \implies p$

(D) It is Sunday is a sufficient condition for the shops to be closed.

(E) The shops are closed is a necessary condition for it to be Sunday.

Question 12. Let $p$ be "You get good marks" and $q$ be "You study hard". The statement "You get good marks only if you study hard" can be represented as:

(A) $p \implies q$

(B) If you get good marks, then you study hard.

(C) $\neg q \implies \neg p$

(D) Studying hard is a necessary condition for getting good marks.

(E) Getting good marks is a sufficient condition for studying hard.

Question 13. Which rows in the truth table make the statement $p \implies q$ true?

p | q | p $\implies$ q

T | T | T

T | F | F

F | T | T

F | F | T

(A) When $p$ is True and $q$ is True.

(B) When $p$ is True and $q$ is False.

(C) When $p$ is False and $q$ is True.

(D) When $p$ is False and $q$ is False.

(E) When the antecedent is false.

Question 14. Which rows in the truth table make the statement $p \iff q$ false?

p | q | p $\iff$ q

T | T | T

T | F | F

F | T | F

F | F | T

(A) When $p$ is True and $q$ is True.

(B) When $p$ is True and $q$ is False.

(C) When $p$ is False and $q$ is True.

(D) When $p$ is False and $q$ is False.

(E) When $p$ and $q$ have different truth values.

Question 15. The statement "A number is divisible by $6$ if and only if it is divisible by both $2$ and $3$". Let $p$ be "A number is divisible by $6$" and $q$ be "A number is divisible by both $2$ and $3$". Which of the following are true?

(A) The statement $p \iff q$ is true for any integer.

(B) $p$ is a sufficient condition for $q$.

(C) $q$ is a necessary condition for $p$.

(D) The statement $p \implies q$ is true.

(E) The statement $q \implies p$ is true.

Question 1. Given the conditional statement "$p \implies q$", which of the following are its related conditional statements?

(A) Converse ($q \implies p$)

(B) Inverse ($\neg p \implies \neg q$)

(C) Contrapositive ($\neg q \implies \neg p$)

(D) Negation ($p \land \neg q$)

(E) Biconditional ($p \iff q$)

Question 2. Which pairs of related conditional statements are logically equivalent?

(A) Conditional ($p \implies q$) and Converse ($q \implies p$)

(B) Conditional ($p \implies q$) and Inverse ($\neg p \implies \neg q$)

(C) Conditional ($p \implies q$) and Contrapositive ($\neg q \implies \neg p$)

(D) Converse ($q \implies p$) and Inverse ($\neg p \implies \neg q$)

(E) Inverse ($\neg p \implies \neg q$) and Contrapositive ($\neg q \implies \neg p$)

Question 3. Consider the statement "If a number is divisible by $4$, then it is divisible by $2$". Let $p$: "a number is divisible by $4$", $q$: "it is divisible by $2$". Which of the following are true statements?

(A) The original statement ($p \implies q$) is true.

(B) The converse ($q \implies p$) is true.

(C) The inverse ($\neg p \implies \neg q$) is true.

(D) The contrapositive ($\neg q \implies \neg p$) is true.

(E) The converse is "If a number is divisible by $2$, then it is divisible by $4$".

Question 4. If the conditional statement $p \implies q$ is false, then which of the following must be true?

(A) $p$ is true.

(B) $q$ is false.

(C) $\neg (p \implies q)$ is true.

(D) $p \land \neg q$ is true.

(E) The contrapositive is false.

Question 5. Write the converse and inverse of the statement "If it is a rainy day (r), then the match is cancelled (c)".

(A) Converse: If the match is cancelled, then it is a rainy day.

(B) Converse: $\neg c \implies \neg r$

(C) Inverse: If it is not a rainy day, then the match is not cancelled.

(D) Inverse: $c \implies r$

(E) Converse: $r \implies c$

Question 6. If the contrapositive of a statement is true, what can you conclude about the original conditional statement?

(A) The original statement is also true.

(B) The original statement is false.

(C) The original statement is logically equivalent to its contrapositive.

(D) The original statement's truth value cannot be determined.

(E) The original statement is logically equivalent to the inverse of its converse.

Question 7. The statement "A polygon is a triangle only if it has three sides". Let $p$: "A polygon is a triangle", $q$: "It has three sides". Which of the following is equivalent to this statement?

(A) If a polygon is a triangle, then it has three sides ($p \implies q$).

(B) If a polygon does not have three sides, then it is not a triangle ($\neg q \implies \neg p$).

(C) A polygon has three sides is a necessary condition for it to be a triangle.

(D) A polygon is a triangle is a sufficient condition for it to have three sides.

(E) If a polygon has three sides, then it is a triangle ($q \implies p$).

Question 8. The negation of "$p \implies q$" is logically equivalent to:

(A) $\neg (\neg p \lor q)$

(B) $p \land \neg q$

(C) $\neg p \land \neg q$

(D) It is not the case that if $p$ then $q$.

(E) $p$ is true and $q$ is false.

Question 9. If the converse of a conditional statement is true, which of the following must also be true?

(A) The original conditional statement.

(B) The inverse of the original conditional statement.

(C) The contrapositive of the original conditional statement.

(D) The negation of the original conditional statement.

(E) The contrapositive of the converse (which is the inverse). ($\neg p \implies \neg q$)

Question 10. Consider the statement "If a quadrilateral is a rhombus, then it is a parallelogram". Which of the following are true statements?

(A) The original statement is true.

(B) The converse is "If a quadrilateral is a parallelogram, then it is a rhombus".

(C) The contrapositive is "If a quadrilateral is not a parallelogram, then it is not a rhombus".

(D) The inverse is "If a quadrilateral is not a rhombus, then it is not a parallelogram".

(E) The converse is true.

Question 11. Which of the following are logically equivalent to the statement "It is not raining or I will go for a walk"? Let $R$: "It is raining", $W$: "I will go for a walk".

(A) $\neg R \lor W$

(B) If it is raining, then I will go for a walk.

(C) $R \implies W$

(D) If I do not go for a walk, then it is not raining.

(E) $\neg W \implies \neg R$

Question 12. Which of the following statements implies $p \implies q$ is true?

(A) $p$ is true and $q$ is false.

(B) $p$ is false and $q$ is true.

(C) $p$ is false and $q$ is false.

(D) $\neg p$ is true.

(E) $q$ is true.

Question 13. If the inverse of a statement is true, which of the following must also be true?

(A) The original conditional statement.

(B) The converse of the original conditional statement.

(C) The contrapositive of the original conditional statement.

(D) The negation of the converse.

(E) The negation of the original conditional statement.

Question 14. Consider the statement "If a number is prime, then it is odd". Which of the following are true about its related statements?

(A) The original statement is false.

(B) The converse is "If a number is odd, then it is prime", which is false.

(C) The inverse is "If a number is not prime, then it is not odd", which is false.

(D) The contrapositive is "If a number is not odd (even), then it is not prime", which is true.

(E) The converse and inverse have the same truth value.

Question 15. If $p \iff q$ is true, which of the following are true?

(A) $p \implies q$ is true.

(B) $q \implies p$ is true.

(C) $\neg p \implies \neg q$ is true.

(D) $\neg q \implies \neg p$ is true.

(E) $p \land \neg q$ is false.

Question 1. Which of the following phrases indicate the use of a universal quantifier ($\forall$)?

(A) For all

(B) There exists

(C) Every

(D) Some

(E) For each

Question 2. Which of the following phrases indicate the use of an existential quantifier ($\exists$)?

(A) For every

(B) There exists

(C) At least one

(D) For some

(E) All

Question 3. Write the negation of the statement "For every real number $x$, $x^2 \geq 0$".

(A) For every real number $x$, $x^2 < 0$.

(B) There exists a real number $x$ such that $x^2 \geq 0$.

(C) There exists a real number $x$ such that $x^2 < 0$.

(D) Not all real numbers $x$ satisfy $x^2 \geq 0$.

(E) Some real number $x$ has $x^2 < 0$.

Question 4. Write the negation of the statement "There exists a student who scored full marks".

(A) There exists a student who did not score full marks.

(B) No student scored full marks.

(C) Every student scored full marks.

(D) All students did not score full marks.

(E) For every student, they did not score full marks.

Question 5. Let $P(x)$ be the statement "$x$ is a prime number" and $Q(x)$ be the statement "$x$ is even". Which of the following correctly symbolise the statement "There exists a prime number that is even"?

(A) $\exists x, P(x) \land Q(x)$

(B) $\exists x, P(x) \implies Q(x)$

(C) Some $x$ such that ($x$ is prime AND $x$ is even).

(D) $\neg (\forall x, \neg (P(x) \land Q(x)))$

(E) $\forall x, P(x) \land Q(x)$

Question 6. Let $P(x)$ be "$x$ is a dog" and $Q(x)$ be "$x$ can bark". Which of the following correctly symbolise the statement "All dogs can bark"?

(A) $\forall x, P(x) \land Q(x)$

(B) $\forall x, P(x) \implies Q(x)$

(C) Every $x$ such that (if $x$ is a dog, then $x$ can bark).

(D) There is no $x$ such that ($x$ is a dog AND $x$ cannot bark).

(E) $\neg (\exists x, P(x) \land \neg Q(x))$

Question 7. The negation of a universally quantified statement is:

(A) An existentially quantified statement.

(B) A universally quantified statement with a negated predicate.

(C) An existentially quantified statement with a negated predicate.

(D) Always false.

(E) Of the form $\neg (\forall x, P(x)) \equiv \exists x, \neg P(x)$.

Question 8. The negation of an existentially quantified statement is:

(A) A universally quantified statement.

(B) An existentially quantified statement with a negated predicate.

(C) A universally quantified statement with a negated predicate.

(D) Always true.

(E) Of the form $\neg (\exists x, P(x)) \equiv \forall x, \neg P(x)$.

Question 9. Which of the following statements are true?

(A) $\forall x \in \mathbb{R}, x^2 \geq 0$.

(B) $\exists x \in \mathbb{Z}, x^2 = 2$.

(C) For every integer $n$, $n+1$ is an integer.

(D) There exists a real number $x$ such that $x+1 = x-1$.

(E) Every natural number is an integer.

Question 10. Which of the following statements are false?

(A) All birds can swim.

(B) Some cats have wings.

(C) For every integer $n$, $n$ is positive.

(D) There exists a triangle with four sides.

(E) No integer is a rational number.

Question 11. The statement "Not all that glitters is gold" is logically equivalent to:

(A) All that glitters is not gold.

(B) Nothing that glitters is gold.

(C) Some glittering things are not gold.

(D) There exists something that glitters and is not gold.

(E) It is false that everything that glitters is gold.

Question 12. The statement "There is no perfect number" is equivalent to:

(A) Some number is perfect.

(B) Every number is not perfect.

(C) For all numbers $x$, $x$ is not perfect.

(D) $\neg (\exists x, P(x))$, where $P(x)$ is "$x$ is a perfect number".

(E) $\forall x, \neg P(x)$, where $P(x)$ is "$x$ is a perfect number".

Question 13. Consider the statement "Some people are honest". Which of the following are logically equivalent to its negation?

(A) Some people are not honest.

(B) All people are honest.

(C) No person is honest.

(D) Every person is dishonest.

(E) For every person $x$, $x$ is not honest.

Question 14. The statement "Every Indian city has at least one railway station". Let $C(x)$ be "$x$ is an Indian city" and $R(y,x)$ be "$y$ is a railway station in city $x$". Which of the following symbolize this statement?

(A) $\forall x, C(x) \implies (\exists y, R(y,x))$

(B) $\exists x, C(x) \land (\exists y, R(y,x))$

(C) For every Indian city $x$, there exists a thing $y$ such that $y$ is a railway station in $x$.

(D) It is not the case that there exists an Indian city without a railway station.

(E) $\neg (\exists x, C(x) \land \neg (\exists y, R(y,x)))$

Question 15. The statement "There is a number that is both rational and irrational" is false. Which of the following are true consequences of this?

(A) Its negation is true.

(B) For every number $x$, $x$ is not both rational and irrational.

(C) For every number $x$, if $x$ is rational, then $x$ is not irrational.

(D) For every number $x$, if $x$ is irrational, then $x$ is not rational.

(E) The set of rational numbers and the set of irrational numbers are disjoint.

Question 1. Which of the following are tautologies?

(A) $p \lor \neg p$

(B) $p \land \neg p$

(C) $p \implies p$

(D) $(p \land q) \implies p$

(E) $(p \implies q) \lor (q \implies p)$

Question 2. Which of the following are contradictions (fallacies)?

(A) $p \land \neg p$

(B) $p \lor \neg p$

(C) $(p \implies q) \land p \land \neg q$

(D) $\neg (p \lor \neg p)$

(E) $p \iff \neg p$

Question 3. A compound statement is a contingency if:

(A) Its truth table column contains only True values.

(B) Its truth table column contains only False values.

(C) Its truth table column contains at least one True and at least one False value.

(D) It is neither a tautology nor a contradiction.

(E) It depends on the specific truth values of its simple propositions.

Question 4. Two statements $P$ and $Q$ are logically equivalent if:

(A) They have the same truth value in all possible cases.

(B) Their truth table columns are identical.

(C) The biconditional $P \iff Q$ is a tautology.

(D) The conditional $P \implies Q$ is a tautology.

(E) Their negations are logically equivalent.

Question 5. Which of the following statements are logically equivalent to $p \implies q$?

(A) $\neg p \lor q$

(B) $q \implies p$

(C) $\neg q \implies \neg p$

(D) $\neg (p \land \neg q)$

(E) If not $q$, then not $p$.

Question 6. Consider the statement $(p \land q) \implies r$. How many rows are needed in its truth table?

(A) 4

(B) 6

(C) 8

(D) $2^3$

(E) It depends on the truth values of $p, q, r$.

Question 7. Which of the following are logically equivalent to $\neg (p \land q)$?

(A) $\neg p \land \neg q$

(B) $\neg p \lor \neg q$

(C) Neither $p$ nor $q$ is true.

(D) It is not the case that both $p$ and $q$ are true.

(E) At least one of $p$ or $q$ is false.

Question 8. Which of the following are logically equivalent to $\neg (p \lor q)$?

(A) $\neg p \lor \neg q$

(B) $\neg p \land \neg q$

(C) Neither $p$ nor $q$ is true.

(D) It is not the case that $p$ or $q$ is true.

(E) Both $p$ and $q$ are false.

Question 9. Consider the statement $(p \lor q) \land \neg p$. Which of the following statements is it logically equivalent to?

(A) $q$

(B) $p \land q$

(C) $\neg p \land q$

(D) $p \lor q$

(E) Contingency

Question 10. Which of the following methods can be used to determine if a compound statement is a tautology, contradiction, or contingency?

(A) Constructing a truth table.

(B) Applying logical equivalences to simplify the statement.

(C) Checking if it contradicts a known tautology.

(D) Checking if its negation is a contradiction.

(E) Checking if its negation is a tautology.

Question 11. Which of the following is an example of a contingency?

(A) The Sun is hot or the Sun is not hot.

(B) The Sun is hot and the Sun is not hot.

(C) If it is raining, then the ground is wet.

(D) $p \implies \neg p$

(E) $(p \land q) \lor r$

Question 12. Consider the statements $P: p \implies q$ and $Q: \neg p \lor q$. Which of the following are true?

(A) $P$ and $Q$ are logically equivalent.

(B) The truth table columns for $P$ and $Q$ are identical.

(C) $P \iff Q$ is a tautology.

(D) $P \land Q$ is a tautology.

(E) $P \lor Q$ is a tautology.

Question 13. Which of the following are properties related to absorption laws?

(A) $p \land (p \lor q) \equiv p$

(B) $p \lor (p \land q) \equiv p$

(C) $p \land (q \lor p) \equiv p$

(D) $p \lor (q \land p) \equiv p$

(E) $p \land (p \land q) \equiv p \land q$

Question 14. Which of the following statements are true regarding tautologies and contradictions?

(A) The negation of a tautology is a contradiction.

(B) The negation of a contradiction is a tautology.

(C) A statement is a tautology if and only if its negation is a contradiction.

(D) The conjunction of a tautology and any statement is a tautology.

(E) The disjunction of a contradiction and any statement is the statement itself.

Question 15. Which of the following pairs of statements are logically equivalent?

(A) $\neg (p \land q)$ and $\neg p \lor \neg q$

(B) $\neg (p \lor q)$ and $\neg p \land \neg q$

(C) $p \implies q$ and $\neg p \lor q$

(D) $p \implies q$ and $q \implies p$

(E) $p \iff q$ and $(p \implies q) \land (q \implies p)$

Question 1. Validating a mathematical statement means establishing its truth. Which of the following are methods used for validating statements?

(A) Direct Proof

(B) Proof by Contrapositive

(C) Proof by Contradiction

(D) Proof by Example

(E) Checking if it is intuitively obvious.

Question 2. In a direct proof of $p \implies q$, you typically:

(A) Assume $p$ is true.

(B) Assume $q$ is false.

(C) Use definitions, axioms, and previously proven theorems.

(D) Deduce that $q$ must be true.

(E) Show that assuming $p$ and $\neg q$ leads to a contradiction.

Question 3. To prove $p \implies q$ using the method of contrapositive, you prove that:

(A) If $q$ is false, then $p$ must be false.

(B) $\neg q \implies \neg p$

(C) Assuming $\neg q$ leads to $\neg p$.

(D) If $p$ is true, then $q$ is true.

(E) The contrapositive is logically equivalent to the original statement.

Question 4. In a proof by contradiction for a statement $P$, you typically:

(A) Assume $P$ is true.

(B) Assume $\neg P$ is true.

(C) Deduce a statement that is always false (a contradiction).

(D) Conclude that the initial assumption ($\neg P$) must be false.

(E) Conclude that $P$ must be true.

Question 5. An argument is valid if:

(A) The premises are true.

(B) The conclusion is true.

(C) It is impossible for all the premises to be true and the conclusion false simultaneously.

(D) The conclusion logically follows from the premises.

(E) The conjunction of the premises implies the conclusion is a tautology.

Question 6. Which of the following are valid argument forms?

(A) Modus Ponens: $(p \implies q) \land p \implies q$

(B) Modus Tollens: $(p \implies q) \land \neg q \implies \neg p$

(C) Hypothetical Syllogism: $(p \implies q) \land (q \implies r) \implies (p \implies r)$

(D) Disjunctive Syllogism: $(p \lor q) \land \neg p \implies q$

(E) Fallacy of the Converse: $(p \implies q) \land q \implies p$

Question 7. Which of the following are invalid argument forms (fallacies)?

(A) Modus Ponens

(B) Modus Tollens

(C) Fallacy of the Converse: $(p \implies q) \land q \implies p$

(D) Fallacy of the Inverse: $(p \implies q) \land \neg p \implies \neg q$

(E) Affirming the Consequent

Question 8. A proof of the statement "The sum of two even integers is even" is typically a direct proof. Which of the following would be steps in such a proof?

(A) Assume two even integers, say $m$ and $n$.

(B) Express $m$ as $2k$ and $n$ as $2j$ for some integers $k$ and $j$.

(C) Consider the sum $m+n = 2k + 2j = 2(k+j)$.

(D) Let $k+j = l$, which is an integer.

(E) Conclude that $m+n = 2l$, which shows the sum is even.

Question 9. To prove the statement "If $n^2$ is even, then $n$ is even" using proof by contrapositive, you would prove which of the following equivalent statements?

(A) If $n^2$ is odd, then $n$ is odd.

(B) $\neg (\text{$n$ is even}) \implies \neg (\text{$n^2$ is even})$

(C) If $n$ is odd, then $n^2$ is odd.

(D) Assuming $n$ is odd leads to $n^2$ being odd.

(E) Assuming $n^2$ is even leads to a contradiction.

Question 10. Which of the following statements are true about proof by contradiction?

(A) It is an indirect proof method.

(B) It is based on the fact that a contradiction is always false.

(C) To prove $P$, you assume $\neg P$ and derive a contradiction $Q \land \neg Q$ for some statement $Q$.

(D) It is used when a direct proof is difficult.

(E) It shows that the negation of the statement is a tautology.

Question 11. An argument form is valid if and only if:

(A) The conjunction of its premises is true.

(B) Its conclusion is true.

(C) The corresponding conditional statement (premises implies conclusion) is a tautology.

(D) It is not possible to have true premises and a false conclusion.

(E) It represents correct logical reasoning.

Question 12. Consider the argument: "If it is Sunday (p), then I will rest (q). I will rest (q). Therefore, it is Sunday (p)." Which of the following are true?

(A) This is an example of Modus Ponens.

(B) This is an example of the Fallacy of the Converse.

(C) This is a valid argument form.

(D) This is an invalid argument form.

(E) If the premises are true, the conclusion is not guaranteed to be true.

Question 13. Consider the argument: "If a number is divisible by 6 (p), then it is divisible by 3 (q). The number is not divisible by 3 ($\neg q$). Therefore, the number is not divisible by 6 ($\neg p$)." Which of the following are true?

(A) This is an example of Modus Tollens.

(B) This is a valid argument form.

(C) This is an example of the Fallacy of the Inverse.

(D) If the premises are true, the conclusion must be true.

(E) This argument form corresponds to the contrapositive.

Question 14. Which of the following statements about a mathematical proof are true?

(A) It is a convincing argument for the truth of a statement.

(B) It must follow logical rules.

(C) It starts with assumptions (premises) and uses valid steps to reach a conclusion.

(D) It provides absolute certainty about the truth of a statement within the given framework.

(E) A single counterexample is enough to disprove a universally quantified statement, but not a proof of its negation.

Question 15. To prove that $\sqrt{2}$ is irrational using contradiction, you would start by assuming $\sqrt{2} = p/q$ where $p, q$ are integers, $q \neq 0$, and $\gcd(p,q) = 1$. What would be typical steps or outcomes in this proof?

(A) Squaring both sides to get $2 = p^2/q^2$, or $2q^2 = p^2$.

(B) Concluding that $p^2$ is even.

(C) Concluding that $p$ must be even, so $p = 2k$ for some integer $k$.

(D) Substituting $p=2k$ into the equation to get $2q^2 = (2k)^2 = 4k^2$, leading to $q^2 = 2k^2$.

(E) Concluding that $q^2$ is even, and thus $q$ is even, which contradicts the assumption that $\gcd(p,q) = 1$.

Question 16. Which of the following statements correctly describe aspects of validating mathematical statements?

(A) Validation is the process of demonstrating the truth of a statement.

(B) For some statements, simple observation is sufficient for validation (e.g., "$2+2=4$").

(C) Complex statements often require formal proofs for validation.

(D) Invalidating a statement requires proving its negation is true.

(E) A statement that cannot be proven true is necessarily false.

Question 17. What does it mean for an argument to be sound?

(A) The argument is valid.

(B) The argument is invalid.

(C) All the premises of the argument are true.

(D) The conclusion of the argument is true.

(E) The argument is valid and all its premises are true.

Question 18. Which of the following are essential components of a formal mathematical proof?

(A) Assumptions (premises or axioms)

(B) Logical deductions based on valid rules of inference.

(C) Reference to definitions and previously established results (theorems).

(D) A clear and unambiguous sequence of steps.

(E) Convincing language, even if the logic is slightly flawed.

Question 19. To show that an argument form is invalid, you need to find a specific case where:

(A) All premises are true and the conclusion is true.

(B) At least one premise is false and the conclusion is false.

(C) All premises are true and the conclusion is false.

(D) The corresponding conditional statement (conjunction of premises implies conclusion) is not a tautology.

(E) The argument leads to a contradiction.

Question 20. Which of the following statements about different proof methods are true?

(A) Direct proof is suitable for statements of the form $p \implies q$ by showing $p$ leads to $q$.

(B) Proof by contrapositive is useful when $\neg q \implies \neg p$ is easier to prove than $p \implies q$.

(C) Proof by contradiction can be used to prove any type of statement, including simple propositions or quantified statements.

(D) Proof by example is a valid method to prove a universally quantified statement.

(E) Proof by exhaustion involves checking every possible case for a finite set of cases.