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# 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