Statements and Logical Connectives

1. What is a Mathematical Statement?

A statement (or proposition) is a sentence that is either true or false, but not both simultaneously. Sentences that are ambiguous, interrogative (questions), exclamatory, or imperative (commands) are not mathematical statements.

• Example of a Statement: "The sum of the angles in a triangle is 180 degrees." (This is a true statement).
• Example of a Statement: "All prime numbers are odd." (This is a false statement, because 2 is a prime number).
• Not a Statement: "Mathematics is fun." (This is a subjective opinion).
• Not a Statement: "Where are you going?" (This is a question).

2. Truth Value

The truth value of a statement is its classification as either True (T) or False (F).

3. Simple and Compound Statements

• Simple Statement: A statement whose truth value does not depend on other statements.
• Compound Statement: A statement formed by combining two or more simple statements using logical connectives.

4. Basic Logical Connectives

A. Negation (NOT)

The negation of a statement $p$ denies the statement $p$. It is denoted by $\sim p$ (read as "not p"). If $p$ is true, then $\sim p$ is false. If $p$ is false, then $\sim p$ is true.

• Example: If $p$: "New Delhi is the capital of India." (True), then $\sim p$: "New Delhi is not the capital of India." (False).

Truth Table for Negation:

$p$	$\sim p$
T	F
F	T

B. Conjunction (AND)

A compound statement formed by connecting two simple statements $p$ and $q$ with the word "and" is called a conjunction. It is denoted by $p \land q$ (read as "p and q"). The statement $p \land q$ is true only if both $p$ and $q$ are true.

• Example: If $p$: "The Earth is a planet." (True) and $q$: "The Sun is a star." (True), then $p \land q$: "The Earth is a planet and the Sun is a star." is True.

Truth Table for Conjunction:

$p$	$q$	$p \land q$
T	T	T
T	F	F
F	T	F
F	F	F

C. Disjunction (OR)

A compound statement formed by connecting two simple statements $p$ and $q$ with the word "or" is called a disjunction. It is denoted by $p \lor q$ (read as "p or q"). The statement $p \lor q$ is false only if both $p$ and $q$ are false. In all other cases, it is true. This is the inclusive OR.

• Example: If $p$: "A square has four sides." (True) and $q$: "A triangle has five sides." (False), then $p \lor q$: "A square has four sides or a triangle has five sides." is True.

Truth Table for Disjunction:

$p$	$q$	$p \lor q$
T	T	T
T	F	T
F	T	T
F	F	F