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| Negation of a Statement: Definition and Symbol (~ or $\neg$) | Writing the Negation of Simple Statements | Compound Statements: Definition and Formation |
Negation and Compound Statements
Negation of a Statement: Definition and Symbol ($\sim$ or $\neg$)
Once we have a mathematical statement (proposition), which is a sentence with a definite truth value, we can apply logical operations to it. The simplest of these operations is negation. Negation allows us to form a new statement that expresses the opposite meaning and has the opposite truth value of the original statement.
Definition of Negation
The negation of a statement $p$ is a statement, denoted by $\sim p$ or $\neg p$, which is true when $p$ is false, and false when $p$ is true. It is the formal logical equivalent of saying "It is not the case that $p$" or simply "not $p$".
Essentially, if a statement $p$ makes a certain claim, its negation $\sim p$ denies that claim.
Symbolic Representation
The negation of a statement $p$ is symbolically represented by:
- $\sim p$ (read as "not p" or "tilde p")
- $\neg p$ (read as "not p")
Both symbols are widely accepted and used interchangeably in logic and mathematics. Choosing one over the other is often a matter of convention within a specific text or course.
Truth Value and Truth Table for Negation
The truth value of the negation $\sim p$ is directly dependent on, and is always the opposite of, the truth value of the original statement $p$. This relationship is fundamental and can be clearly illustrated using a truth table.
A truth table shows all possible truth values of a statement(s) and the resulting truth value of a logical operation applied to it/them.
For negation, which operates on a single statement $p$, the truth table has two rows representing the two possible truth values for $p$:
| $p$ | $\sim p$ (or $\neg p$) |
|---|---|
| T (True) | F (False) |
| F (False) | T (True) |
This table formally defines the negation operator: if $p$ is True, $\sim p$ is False, and if $p$ is False, $\sim p$ is True. Negation is a unary operation because it takes only one input statement.
Competitive Exam Focus: Negation Basics
Understanding the concept and symbol of negation is foundational. You should be able to:
- Define negation in your own words.
- Recognise the symbols $\sim$ and $\neg$ as representing negation.
- Instantly recall or reconstruct the truth table for negation.
- Determine the truth value of $\sim p$ if you know the truth value of $p$, and vice-versa. For instance, if you are told $\sim q$ is False, you know that $q$ must be True.
This basic understanding is essential before moving on to negating actual sentences and understanding how negation interacts with other logical connectives.
Writing the Negation of Simple Statements
Forming the negation of a simple declarative statement involves constructing a new sentence that accurately conveys the opposite meaning of the original statement. The goal is that the resulting sentence is true precisely when the original statement is false, and false when the original is true.
Methods for Writing Negation
For simple statements, the negation can often be formed using one of the following approaches:
- Inserting "not": This is the most common method. Place the word "not" or the contraction "-n't" appropriately within the sentence to reverse its meaning. This might require changing the verb form (e.g., "is" to "is not", "does" to "does not").
- Using "It is false that...": Prefix the original statement with the phrase "It is false that". This method is straightforward and works for any statement, although the resulting sentence can sometimes sound awkward or overly formal.
- Using "It is not the case that...": Similar to the previous method, prefix the original statement with "It is not the case that". This also works universally but can feel wordier.
- Using Antonyms or Opposite Phrases: Sometimes, using a word with an opposite meaning can form a natural negation (e.g., "is large" becomes "is small" - although be cautious as this isn't always a direct logical negation unless "small" is defined as "not large"). For mathematical inequalities, symbols like $=$ becomes $\neq$, $<$ becomes $\geq$, and $>$ becomes $\leq$.
The key is to ensure the negation has the precise opposite truth value in all circumstances compared to the original statement.
Example 1. Write the negation of the following statements:
(i) $p$: Paris is in France.
(ii) $q$: $3 + 4 = 7$.
(iii) $r$: The number 7 is a prime number.
(iv) $s$: All dogs are mammals.
(v) $t$: Some birds can fly.
(vi) $u$: The integer $k$ is greater than 10.
Answer:
(i) Statement $p$: Paris is in France.
Truth Value of $p$: True (Based on geography).
Negation $\sim p$:
- Using "not": Paris is not in France.
- Using "It is false that": It is false that Paris is in France.
Let's check the truth value of $\sim p$: Since $p$ is True, $\sim p$ must be False. "Paris is not in France" is indeed False.
(ii) Statement $q$: $3 + 4 = 7$.
Truth Value of $q$: True (Based on arithmetic).
Negation $\sim q$:
- Using "not" (with mathematical symbol): $3 + 4 \neq 7$.
- Using "It is false that": It is false that $3 + 4 = 7$.
Let's check the truth value of $\sim q$: Since $q$ is True, $\sim q$ must be False. "$3 + 4 \neq 7$" is indeed False.
(iii) Statement $r$: The number 7 is a prime number.
Truth Value of $r$: True (Based on the definition of a prime number - 7's only positive divisors are 1 and 7).
Negation $\sim r$:
- Using "not": The number 7 is not a prime number.
- Using "It is false that": It is false that the number 7 is a prime number.
Let's check the truth value of $\sim r$: Since $r$ is True, $\sim r$ must be False. "The number 7 is not a prime number" is indeed False.
(iv) Statement $s$: All dogs are mammals.
Truth Value of $s$: True (Based on biological classification).
Negation $\sim s$:
Negating statements with "all" requires care. The opposite of "All A are B" is NOT "All A are not B". It is "Not all A are B" or "Some A are not B" or "There exists at least one A that is not B".
- Correct Negation: Not all dogs are mammals. OR Some dogs are not mammals. OR There exists at least one dog that is not a mammal.
- Using "It is false that": It is false that all dogs are mammals.
Let's check the truth value of $\sim s$: Since $s$ is True, $\sim s$ must be False. "Some dogs are not mammals" is False because all dogs are indeed mammals.
(v) Statement $t$: Some birds can fly.
Truth Value of $t$: True (Based on observation - many birds like sparrows, crows, eagles can fly. The word "Some" means "at least one").
Negation $\sim t$:
Negating statements with "some" also requires care. The opposite of "Some A are B" is "No A are B" or "All A are not B".
- Correct Negation: No birds can fly. OR All birds cannot fly. OR It is false that there exists a bird that can fly.
- Using "It is false that": It is false that some birds can fly.
Let's check the truth value of $\sim t$: Since $t$ is True, $\sim t$ must be False. "No birds can fly" is False because we know there are birds that *can* fly.
(vi) Statement $u$: The integer $k$ is greater than 10.
Truth Value of $u$: Ambiguous (Depends on the value of $k$. This is an open sentence, not a statement in isolation).
However, if treated as a property we *can* negate it. The negation of "$k > 10$" is the condition that "$k$ is not greater than 10".
Negation of the condition "$k > 10$":
- Using inequality: $k \leq 10$.
- In words: The integer $k$ is less than or equal to 10.
If the original sentence was a statement like "For a specific integer $k=5$, $k > 10$", its negation would be "For the specific integer $k=5$, $k \not> 10$" or "$k \leq 10$". The original statement is false, its negation is true ($5 \leq 10$).
Important Note on Quantifiers: As seen in examples (iv) and (v), negating statements involving quantifiers like "all" ($\forall$) or "some" (or "there exists") ($\exists$) follows specific rules, which we will cover in more detail in the section on Quantifiers.
- $\sim (\text{All P are Q}) \equiv \text{Some P are not Q}$ ($\sim (\forall x, P(x)) \equiv \exists x, \sim P(x)$)
- $\sim (\text{Some P are Q}) \equiv \text{No P are Q} \equiv \text{All P are not Q}$ ($\sim (\exists x, P(x)) \equiv \forall x, \sim P(x)$)
Competitive Exam Pointer: Negating Sentences
This is a frequent topic in logic-based questions. Pay special attention to:
- Negating inequalities: The negation of $a < b$ is $a \geq b$, the negation of $a > b$ is $a \leq b$, the negation of $a = b$ is $a \neq b$.
- Negating quantifiers ("All", "Some", "No"): This is critical. Learn the equivalence rules:
- "All... are..." $\implies$ "Some... are not..."
- "Some... are..." $\implies$ "No... are..." (or "All... are not...")
- "No... are..." $\implies$ "Some... are..."
- "Some... are not..." $\implies$ "All... are..."
- Ensure the negated sentence is grammatically correct and reads naturally if possible, but prioritize logical correctness (opposite truth value). The "It is false that..." construction is a safe bet if unsure about natural phrasing.
Compound Statements: Definition and Formation
Simple statements, while foundational, are rarely sufficient to represent the complexity of arguments in mathematics or everyday reasoning. We need ways to combine these simple statements to form more intricate logical structures. Statements formed by joining two or more simple statements are called compound statements.
Definition of a Compound Statement
A compound statement is a statement constructed by combining two or more simple statements using logical connectors. The truth value of a compound statement depends entirely on the truth values of its constituent simple statements and the specific type of connector used.
Component Statements
The simple statements that are combined to form a compound statement are referred to as its component statements or sometimes constituent statements or atomic statements. Identifying these components is the first step in analyzing the logic of a compound statement.
Logical Connectives (Operators)
The words or phrases used to combine simple statements into compound statements are called logical connectives or logical operators. These connectives have precise definitions in terms of how they affect the truth value of the resulting compound statement based on the truth values of the components.
The most common logical connectives are:
- Conjunction ("and"): Combines two statements to assert that both are true. Symbol: $\land$. Read as "$p$ and $q$".
- Disjunction ("or"): Combines two statements to assert that at least one of them is true. Symbol: $\lor$. Read as "$p$ or $q$".
- Conditional ("if... then..."): Connects two statements to express an implication. Symbol: $\implies$ or $\rightarrow$. Read as "If $p$, then $q$". $p$ is the hypothesis/antecedent, $q$ is the conclusion/consequent.
- Biconditional ("if and only if"): Connects two statements to express that they have the same truth value. Symbol: $\iff$ or $\leftrightarrow$. Read as "$p$ if and only if $q$". Often abbreviated as "iff".
Negation ("not", $\sim$ or $\neg$) is also sometimes listed as a connective, although it is technically a unary operator (operating on one statement) rather than a binary connective (operating on two statements).
We use lowercase letters like $p, q, r, s, ...$ to represent simple statements.
Example 1. For each of the following compound statements, identify the component statements and the logical connective used.
(i) The sun is shining and it is warm.
(ii) I will go to the market or I will read a book.
(iii) If you study hard, then you will pass the exam.
(iv) A quadrilateral is a square if and only if it is a rhombus and it is a rectangle.
(v) It is not raining.
Answer:
(i) Statement: "The sun is shining and it is warm."
- Component Statement 1 ($p$): The sun is shining.
- Component Statement 2 ($q$): It is warm.
- Connective: and ($\land$).
- Symbolic form: $p \land q$.
(ii) Statement: "I will go to the market or I will read a book."
- Component Statement 1 ($p$): I will go to the market.
- Component Statement 2 ($q$): I will read a book.
- Connective: or ($\lor$).
- Symbolic form: $p \lor q$.
(iii) Statement: "If you study hard, then you will pass the exam."
- Component Statement 1 ($p$): You study hard. (This is the part following "If")
- Component Statement 2 ($q$): You will pass the exam. (This is the part following "then")
- Connective: if... then... ($\implies$).
- Symbolic form: $p \implies q$.
(iv) Statement: "A quadrilateral is a square if and only if it is a rhombus and it is a rectangle."
This statement is more complex as one of the components of the biconditional is itself a compound statement.
- Main Component Statement 1 ($p$): A quadrilateral is a square.
- Main Connective: if and only if ($\iff$).
- Main Component Statement 2 ($q$): It is a rhombus and it is a rectangle.
- Now, let's break down $q$:
- Sub-component Statement 1 ($r$): It (the quadrilateral) is a rhombus.
- Sub-component Statement 2 ($s$): It (the quadrilateral) is a rectangle.
- Connective within $q$: and ($\land$).
- Symbolic form of $q$: $r \land s$.
- Overall Symbolic form: $p \iff (r \land s)$.
(v) Statement: "It is not raining."
This statement involves negation.
- Original Simple Statement ($p$): It is raining.
- Connective/Operator: not ($\sim$ or $\neg$).
- Symbolic form: $\sim p$ (or $\neg p$).
While technically negation is a unary operator and the result is a statement rather than a combination of two statements, sentences formed using "not" are often considered alongside compound statements because negation modifies the truth value of a statement.
The ability to dissect compound statements into their components and identify the connectives is fundamental to constructing truth tables and analyzing the logical structure of arguments, which will be explored in subsequent sections.
Competitive Exam Pointer: Identifying Components and Connectives
In exams, you might be asked to:
- Identify the component statements in a given compound statement.
- Identify the logical connective used.
- Translate a statement from English into symbolic form using statement variables ($p, q, ...$) and logical symbols ($\land, \lor, \implies, \iff, \sim$).
- Recognise that complex statements can have multiple levels of compounding, as shown in Example 1 (iv).
Paying close attention to words like "and", "or", "if... then...", "if and only if", and "not" is key to correctly identifying the structure of a compound statement.