Logical equality

Logical equality

EQ, XNOR
Definition	${\displaystyle x=y}$
Truth table	${\displaystyle (1001)}$
Logic gate
Normal forms
Disjunctive	${\displaystyle x\cdot y+{\overline {x}}\cdot {\overline {y}}}$
Conjunctive	${\displaystyle ({\overline {x}}+y)\cdot (x+{\overline {y}})}$
Zhegalkin polynomial	${\displaystyle 1\oplus x\oplus y}$
Post's lattices
0-preserving	no
1-preserving	yes
Monotone	no
Affine	yes
Self-dual	no
v t e

Logical equality is a logical operator that corresponds to equality in Boolean algebra and to the logical biconditional in propositional calculus. It gives the functional value true if both functional arguments have the same logical value, and false if they are different.

It is customary practice in various applications, if not always technically precise, to indicate the operation of logical equality on the logical operands x and y by any of the following forms:

${\displaystyle {\begin{aligned}x&\leftrightarrow y&x&\Leftrightarrow y&\mathrm {E} xy\\x&\mathrm {~EQ~} y&x&=y\end{aligned}}}$

Some logicians, however, draw a firm distinction between a functional form, like those in the left column, which they interpret as an application of a function to a pair of arguments — and thus a mere indication that the value of the compound expression depends on the values of the component expressions — and an equational form, like those in the right column, which they interpret as an assertion that the arguments have equal values, in other words, that the functional value of the compound expression is true.