Logical Equivalence

The equivalence of two statements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

$P$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \iff Q}
True	True	True
True	False	False
False	True	False
False	False	True

The logical symbol for implication is "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iff} ", so you can write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \iff Q} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} iff Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} .

The statement

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} iff Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q}

states that the implication

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q}

and its converse are both true.

Resources

• Logical Equivalence, Wikipedia
• Logical Equivalences, Mathematics LibreTexts
• Logical Equivalence and Truth Tables, United States Naval Academy College of Mathematics

Licensing

Content obtained and/or adapted from:

• Logical Connectives, Wikibooks: Mathematical Proof and Principles of Mathematics under a CC BY-SA license