Logic Sets and Real Number System

Logic and Statement

What is Logic?

Logic is the study of reasoning. It helps us determine whether a statement is true or false.

Statement: 

A statement (or proposition) is a declarative sentence that is either true or false, but not both at the same time.  It cannot be a question, command, or opinion.

Examples: 

- “Kathmandu is the capital of Nepal.” → True 

- “2 + 2 = 5.” → False 

- “x + 2 = 7” → This is not a statement (it depends on x; called an open sentence). 

Note: Statements are usually denoted by small letters: \( p, q, r \).

Truth Value and Truth Table

Truth Value:

The truth of the falsity of a statement is known as its truth value. Every statement has exactly one truth value:  T (True), F (False).

Truth Table:

The table presenting the truth values of the component statements together with the truth values of their compound statement is called the truth table. A truth table lists all possible combinations of truth values of the given statements and shows the truth value of the resulting compound statement.

Logical Connectives

The compound statements are made up of simple statements. The simple statements are connected by the words like 'and', 'or', 'if … then' and 'if and only if'. The connecting phrases are called logical connectives or simply connectives. 

1. Conjunction (and)

2. Disjunction (or)

3. Negation (not)

4. Implication (If .. then)

5. Bidirectional (iff)