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 compares two truth values, or more generally, two formulas, such that it gives the value True if both arguments have the same truth value, and False if they are different. In the case where formulas have free variables, we say two formulas are equal when their truth values are equal for all possible resolutions of free variables. It corresponds to equality in Boolean algebra and to the logical biconditional in propositional calculus.

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.^{[citation needed]}