Logic

The field of logic consideres truth values of mathematical expressions.

Logical Operators

Truth values can be transformed or combined by logical operators.

Not

The \(\mathrm{not}\) operator inverts the truth values of an expression.

\(a\)	\(\mathrm{not}\,a\)
true	false
false	true

And

The \(\mathrm{and}\) operator yields true if and only if both operands are true.

\(a\)	\(b\)	\(a\,\mathrm{and}\,b\)
true	true	true
true	false	false
false	true	false
false	false	false

Or (inclusive)

The \(\mathrm{or}\) operator yields true if and only if at least one of both operands is true. This is sometimes called inclusive or because the and-case (both operands true) is included (that is, yields true).

\(a\)	\(b\)	\(a\,\mathrm{or}\,b\)
true	true	true
true	false	true
false	true	true
false	false	false

Or (exclusive)

The \(\mathrm{xor}\) operator yields true if and only if exactly one of both operands is true. This called exclusive or because the and-case is excluded.

\(a\)	\(b\)	\(a\,\mathrm{xor}\,b\)
true	true	false
true	false	true
false	true	true
false	false	false