Truth Table

Definition

A truth table lists all possible truth value combinations for propositions.

It shows the resulting truth value of a compound proposition.

Example:

P      Q      P AND Q
true   true   true
true   false  false
false  true   false
false  false  false

A truth table makes a logical operator's behavior explicit.

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Why Truth Tables Matter

Logical operators can be confusing if described only in words.

A truth table removes ambiguity.

Example:

P OR Q

Truth table:

P      Q      P OR Q
true   true   true
true   false  true
false  true   true
false  false  false

This shows that logical OR is true when both parts are true.

Truth tables reveal the exact truth pattern.

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Rows Represent Cases

Each row is one possible case.

For two propositions:

P
Q

there are four combinations:

true, true
true, false
false, true
false, false

The table checks the expression under each case.

A truth table tests every possible truth assignment.

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Truth Table For NOT

NOT has one input proposition.

P      NOT P
true   false
false  true

This shows that NOT reverses truth value.

A truth table can describe one-input or multi-input operators.

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Truth Tables For Compound Expressions

Truth tables can evaluate larger expressions.

Example:

(P AND Q) OR R

To evaluate it, break it into parts:

P AND Q
then
(P AND Q) OR R

Each intermediate column helps compute the final result.

Complex expressions can be evaluated step by step.

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Final Summary

• A truth table lists possible truth assignments.
• It shows the truth value of a compound proposition.
• Each row represents one possible case.
• Truth tables make logical operators precise.
• They can evaluate simple or complex expressions.
• Truth tables are a practical tool for checking symbolic reasoning.