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Truth Tables and Logical Equivalence

Computer Science I 235 words Free to read

Truth Tables and Equivalence

A truth table lists the output of a boolean function for every possible input combination. For nn inputs, the table has 2n2^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 (nn)Rows (2n2^n)
24
38

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:

LawEquationsDescription
IdentityA+0=AA + 0 = A, A⋅1=AA \cdot 1 = ANeutral elements
ComplementA+A‾=1A + \overline{A} = 1, A⋅A‾=0A \cdot \overline{A} = 0Inverse logic
De MorganA⋅B‾=A‾+B‾\overline{A \cdot B} = \overline{A} + \overline{B}Negation flips operator
De MorganA+B‾=A‾⋅B‾\overline{A + B} = \overline{A} \cdot \overline{B}Negation flips operator
Common pitfall: Applying De Morgan's law incorrectly by writing A⋅B‾=A‾⋅B‾\overline{A \cdot B} = \overline{A} \cdot \overline{B} (wrong) instead of A‾+B‾\overline{A} + \overline{B} (right). Negation always flips the operator.
De Morgan's law checked row by row -- and the flipped-operator pitfall

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Basic Digital Design