# Propositional Logic Introduction

Boolean Algebra

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This is an introduction to Propositional Logic tutorial.

## What is a Proposition?

A Proposition is an atomic sentence that can either be TRUE or FALSE and nothing else.

Following sentences are example of proposition.

Proposition: India is a country
Result: TRUE

Proposition: 100 is greater than 200
Result: FALSE

Whereas the sentence How are you? is not a proposition as the answer can’t be TRUE or FALSE.

## Simple and Compound Proposition

A simple proposition is one that does not contain any other propositions as its part.

A compound proposition is one that is made up of two or more simple propositions.

We use lower case letters a,b,c… to represent proposition.

## Example of simple proposition

It is raining.

This statement can either be TRUE or FALSE.

## Example of compound proposition

Today is Sunday and Sunday is a holiday.

This statement contain two simple propositions "Today is Sunday" and "Sunday is a holiday" both the statement can be either TRUE or FALSE.

## Operator or Logical Connective

Operator or logical connective are the things that joins simple propositions into compound propositions and joins compound propositions into larger compound propositions.

## What is Propositional Logic?

Propositional Logic is a way to represent logic through propositions and logical connectives.

## Types of Logical Connectives (Operators)

Following are the types of logical connectives (operators) used in propositional logic.

• Disjunctive (also called OR)
• Conjunctive (also called AND)
• Conditional (also called Implication)
• Bi-conditional (also called Equivalence)
• Negation (also called NOT)

## Disjunctive operator

Disjunctive (also called OR) means one of the two arguments is true or both of them are true.

We use the word OR and + and ∨ symbols to represent disjunctive.

Example
p + q
p ∨ q
p OR q
They all mean
either p is true, or q is true, or both are true.

Consider two arguments (proposition)
p = Oct 21, 2012 was Sunday
q = Sunday is a holiday
then,
p + q
p ∨ q
p OR q

They all means either p is true, or q is true, or both are true.

i.e., either Oct 21, 2012 was Sunday is true or Sunday is a holiday is true or both are true.

## Conjunctive operator

Conjunctive (also called AND) means both the arguments are true. We use the word AND and . & and ∧ symbols to represent conjunctive.

Example
p . q
p & q
p ∧ q
p AND q

They all means both p and q are true.

Consider two arguments (proposition)
p = Oct 21, 2012 was Sunday
q = Sunday is a holiday
then,
p . q
p & q
p ∧ q
p AND q

They all means both p and q are true.

i.e., both Oct 21, 2012 was Sunday and Sunday is a holiday are true.

## Conditional (Implication) operator

Conditional also called Implication (If...Then).

Implication means if one argument is true then the other argument is true.

We use the ⇒ symbol to represent conditional operator.

Example
p ⇒ q
this means if p is true, then q is true.

Consider two arguments (proposition)
p = 10 is greater than 0
q = 10 is positive
then,
p ⇒ q
this means if p is true, then q is true.
i.e., if 10 is greater than 0 then 10 is positive.

## Bi-conditional (Equivalence) operator

Bi-conditional also called Equivalence (If and only If).

Equivalence means either both arguments are true or both are false.

We use the ⇔ symbol to represent bi-conditional.

Example
p ⇔ q
this means
p and q either both are true or both are false.

Consider two arguments (proposition)
p = 10 is greater than 0
q = 10 is positive
then,
p ⇔ q
this means
p and q either both are true or both are false.

i.e., 10 is greater than 0 and 10 is positive
either both are true or both are false.

## Negation (also called NOT)

Negation is an operator that affects only one statement and does not join two statements.

We use the ~ and ' symbol to represent negation

~p
p'
this means
if p is true then,
~p is false.

Example

Consider the argument (proposition)
p = It is raining
if p is true i.e., it is raining
then,
~p is false i.e., it is not raining.

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