In logics and mathematics, negation (from Latin negare ‘to deny’) is the unary operation “” which swaps the truth value of any operand to the opposite truth value. So, if the statement is true then its negated statement is false, and vice versa.
Note 1. The negated statement (by Heyting) has been denoted also with (Peano), (Russell), (Hilbert) and (by the Polish notation).
Note 2. may be expressed by implication as
where means any contradictory statement.
Note 3. The negation of logical or and logical and give the results
Analogical results concern the quantifier statements:
These all are known as de Morgan’s laws.
Note 4. Many mathematical relation statements, expressed with such special relation symbols as , are negated by using in the symbol an additional cross line: .