Prepositional Logic – Logical Connectives and Truth Tables

Introduction

We learned about the statements which has a truth value. A statement can be True or False, but not both. You can make new statements from simple statements, such a statement is called a Compound Preposition or a Compound Statement.

For example,

Let  p and q be two statements.

p:  The triangle has three equal sides.
q: The triangle is an equilateral triangle.

The compound statement is a statement using these two statements.

p ∧ q: The triangle has three equal sides and it is an equilateral triangle.
p ∨ q: The triangle has three equal sides and it is an equilateral triangle.

We constructed the compound statements using simple statements by using a connective. A connective is a logical symbol that connects two simple statements. The are many connectives, they are listed below.

NameSymbol
and\wedge
or\vee
implications\Rightarrow
bi-conditionals\Leftrightarrow

Note that there are rule associated with each of the logical connectives and we will discuss about them.

Truth Table for Compound Statements

Now that we know how to create a compound statement, we find that there are more than one variable working on it.

We want to know the final truth value of the compound preposition which only possible when we verify all possible combination of there individual variables.

This listing can be described in a tabular format called the truth table. Another main purpose of the truth table is to show the logical equivalence of two compound statements.

For Example

We have given the truth table for following

p   ∨   q 

Rule: If any of the variable p , q is True, then the statement is True, else it is false.

p  ∧  q

Rule: When  both the variable p , q are True, then the statement is True, otherwise it is false.

p ⇒ q

Rule: The only  case when the statement is False is when p is True and q is False, where p implies q Otherwise, the statement is True.

p ⇔  q

Rule: When both the variables p , q  are same then the statement is  True, Otherwise , it is False.
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