Logical Equivalence

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

Some ways to phrase this are

$P$ is equivalent to $Q$.

$P$ if and only if $Q$.

$P$ exactly when $Q$.

$P$ iff $Q$. (iff is an abbreviation for if and only if).

$P$ is a necessary and sufficient condition for $Q$.

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

$P$	$Q$	$P\iff Q$
True	True	True
True	False	False
False	True	False
False	False	True

The logical symbol for implication is "$\iff$", so you can write $P\iff Q$ for $P$ iff $Q$.

The statement

$P$ iff $Q$

states that the implication

$P$ implies $Q$

and its converse are both true.

Resources

• Logical Connectives, Wikibooks: Mathematical Proof and Principles of Mathematics
• Logical Equivalence, Wikipedia
• Logical Equivalences, Mathematics LibreTexts
• Logical Equivalence and Truth Tables, United States Naval Academy College of Mathematics