is the symbol for the Boolean value false.

Background & Context

  • False is the symbol that represents the Boolean value false. Expressions that can be rigorously established to be false return this symbol. Examples of testing expressions that may return False include Equal, Unequal, SameQ, UnsameQ, Less/Greater/etc., Exists, and quantifier elimination via Resolve. While "Q"-functions (e.g. TrueQ, SameQ, UnsameQ) always return True or False, non-Q comparison and equality-testing functions (e.g. Equal, Unequal, Less, Greater) return unevaluated when they cannot be definitively resolved. Constructs that can be used to take a different evaluation path depending on if a condition is False or True include If, Which, and Piecewise.
  • The negation Not[False] of False is given by True. The domain consisting of False and True is denoted Booleans.
  • While TrueQ is a special case of If that yields True if an expression is explicitly True and otherwise yields False, there is no corresponding built-in function for False.


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Basic Examples  (4)

Evaluate a Boolean expression:

Use a conditional:

Test a structural property:

Test a mathematical property:

Properties & Relations  (6)

The symbol for the Boolean value true:

Truth table for a Boolean function:

The Boole function:

This statement is not resolved automatically:

Use FullSimplify to find its truth value:

A fully quantified expression:

Use Resolve to find its truth value:

Use Refine to find truth values of expressions under specified assumptions:

Introduced in 1988