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LOGIC AND PROOFS- Truth Tables

A truth table is a systematic way to show all possible combinations of input values and their corresponding output values for logical operations or expressions.

Basic Structure

A truth table has columns for:

  • Each input variable (usually labeled with letters like A, B, C)
  • The logical operation or expression being evaluated
  • The resulting output (typically True/False or 1/0)

Common Logical Operations

NOT (¬)

A¬ATFFT

AND (∧)

ABA ∧ BTTTTFFFTFFFF

OR (∨)

ABA ∨ BTTTTFTFTTFFF

XOR (⊕) – Exclusive OR

ABA ⊕ BTTFTFTFTTFFF

IMPLIES (→)

ABA → BTTTTFFFTTFFT

Creating Truth Tables

For complex expressions with n variables, you need 2ⁿ rows to cover all possibilities. The key steps are:

  1. List all possible combinations of input values
  2. Work through the expression step by step
  3. Calculate intermediate results if needed
  4. Determine the final output for each combination

Truth tables are fundamental tools in logic, computer science, digital circuit design, and mathematical proofs. They help verify logical equivalences, identify tautologies and

Truth Table Calculator

Generate truth tables for logical expressions

Supported Operators:

&&
AND
||
OR
!
NOT
^
XOR
->
IMPLIES
<->
BICONDITIONAL

Example Expressions:

  • (A && B) || C – A and B, or C
  • !(A || B) – Not (A or B)
  • A -> B – If A then B
  • A <-> B – A if and only if B
  • A ^ B ^ C – XOR of A, B, and C