Truth Tables and Logical Equivalence

Truth Tables and Equivalence

A truth table lists the output of a boolean function for every possible input combination. For $n$ inputs, the table has $2^n$ rows. It provides a complete specification of a function.

Two expressions have logical equivalence if and only if their truth tables match row for row. This gives a foolproof test for equality no matter how different the expressions look.

Inputs ( $n$ )	Rows ( $2^n$ )
2	4
3	8

Each row where output is 1 is a minterm. OR-ing minterms together creates a canonical sum-of-products expression, proving any function can be built from gates.

Boolean Laws and Pitfalls

Truth tables verify the laws of boolean algebra:

Law	Equations	Description
Identity	$A + 0 = A$ , $A \cdot 1 = A$	Neutral elements
Complement	$A + \overline{A} = 1$ , $A \cdot \overline{A} = 0$	Inverse logic
De Morgan	$\overline{A \cdot B} = \overline{A} + \overline{B}$	Negation flips operator
De Morgan	$\overline{A + B} = \overline{A} \cdot \overline{B}$	Negation flips operator

Common pitfall: Applying De Morgan's law incorrectly by writing $\overline{A \cdot B} = \overline{A} \cdot \overline{B}$ (wrong) instead of $\overline{A} + \overline{B}$ (right). Negation always flips the operator.

De Morgan's law checked row by row -- and the flipped-operator pitfall

Truth Tables and Equivalence

Boolean Laws and Pitfalls

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