Negation of Statements

The branch of mathematics that deals with reasoning principles is termed mathematical logic. A sentence that is declarative in nature, whose validity is either true or false, has to be decided, is called a logical sentence. They are usually denoted as p, q and r. The types of logical statements are as follows:

1) A statement consisting of 1 or more variables, which are given values, becomes an open statement.

2) A compound statement is defined as the combination of two or more simple statements using words such as “and”, “or”, “not”, “if”, “then”, and “if and only if”.

A given statement can be either true or false, which are the truth values of the statement. They are denoted as T and F. A truth table is one that consists of all the truth values of the statements for all different combinations of values assigned to the variables of the compound statement.

Logical Operations

Logical operations or logical connectives are those that connect simple statements.

The different kinds of logical operations are discussed below.

a) AND operation: A statement that is compound in nature, formed by 2 simple sentences (say p and q) using a logical connective, is termed a conjunction of p and q. It is given by p ∧ q.

b) OR operation: A statement that is compound in nature, formed by 2 simple sentences (say p and q) using a logical connective, is termed a disjunction of p and q. It is given by p ∨ q.

c) Conditional Operation: 2 simple statements that are connected by the phrase “if and then” is called a conditional statement. It is given by p ⇒ q.

d] Biconditional Operation: 2 simple statements that are connected by the phrase “if and only if” are called biconditional statements.  They are given by the symbol ⇔.

e] Negation/NOT Operation: A statement that is constructed by interchanging the truth value of the statement is called the negation of that statement. It is done by using words like “no”, “not” and denoted as ~ (statement). It is called a logical connective even if it doesn’t connect two statements.

Negation of compound sentences is as follows:

Consider p and q to be two statements.

(i) Negation of conjunction: ~ (p ∧ q) ≡ (~ p ∨ ~ q)

(ii) Negation of disjunction: ~ (p ∨ q) ≡ (~ p ∧ ~ q)

(iii) Negation of implication: ~ (p ⇒ q) = (p ∧ ~ q)

(iv) Negation of biconditional statement or equivalence: ~ (p ⇔ q) = (p ∧ ~ q) ∨ (q ∧ ~ p)

Example 1: Let S be a non-empty subset of R. Consider the following statement: P: There is a rational number x ∈ S, such that x > 0. Which of the following statements is the negation of the statement p?

A) x ∈ S and x ≤ 0 ⇒ x is not rational.

B) There is a rational number x ∈ S, such that x ≤ 0.

C) There is no rational number x ∈ S, such that x ≤ 0.

D) Every rational number x ∈ S satisfies x ≤ 0.

Solution: D

~ P: Every rational number x ∈ S satisfies x ≤ 0.

Example 2: Consider the following statements:

P: Suman is brilliant.

Q: Suman is rich.

R: Suman is honest.

The negation of the statement –

“Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as

Solution:

“Suman is brilliant and dishonest” can be expressed as P ∧ ∼ R

So, “Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as (P ∧ ∼ R) ↔ Q.

Now, negation of this = ∼ [(P ∧ ∼ R) ↔ Q]

p ↔ q ≡ q ↔ p

∼ [(P ∧ ∼ R) ↔ Q] ≡ ∼ [Q ↔ (P ∧ ∼ R)]

Negation of Statements – Video Lesson