Logical Equivalence

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The equivalence of two statements and is the statement is that and have the same truth value. Another way of say this is that implies and implies .

Some ways to phrase this are

is equivalent to .
if and only if .
exactly when .
iff . (iff is an abbreviation for if and only if).
is a necessary and sufficient condition for .

Examples:

First statement Second statement Equivalence
4 is even. 6 is odd. 4 is even iff 6 is odd.
Triangle ABC is equilateral. Triangle ABC is equiangular. Triangle ABC is equilateral exactly when it is equiangular.

The equivalence iff is True when and have the same truth values, and False when they have different truth values. In other words iff is True when and are both True or both False, and iff is False is one of and is True while the other is false. In tabular form:

True True True
True False False
False True False
False False True

The logical symbol for implication is "", so you can write for iff .

The statement

iff

states that the implication

implies

and its converse are both true.

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