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

.

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]

Logical equality is an

, that produces a value of true if and only if both operands are false or both operands are true.

The truth table of p EQ q (also written as p = q, p ↔ q, Epq, p ≡ q, or p == q) is as follows:

Logical equality

p	q	p = q
0	0	1
0	1	0
1	0	0
1	1	1

The form (x = y) is equivalent to the form (x ∧ y) ∨ (¬x ∧ ¬y).

${\displaystyle (x=y)=\lnot (x\oplus y)=\lnot x\oplus y=x\oplus \lnot y=(x\land y)\lor (\lnot x\land \lnot y)=(\lnot x\lor y)\land (x\lor \lnot y)}$

For the operands x and y, the truth table of the logical equality operator is as follows:

${\displaystyle x\leftrightarrow y}$		y
		T	F
x	T	T	F
	F	F	T

In

signified by "XOR" or "⊕". Naturally, these variations in usage have caused some failures to communicate between mathematicians and switching engineers over the years. At any rate, one has the following array of corresponding forms for the symbols associated with logical inequality:

${\displaystyle {\begin{aligned}x&+y&x&\not \equiv y&Jxy\\x&\mathrm {~XOR~} y&x&\neq y\end{aligned}}}$

This explains why "EQ" is often called "

Another rationalization of the admittedly circuitous name "XNOR" is that one begins with the "both false" operator NOR and then adds the eXception "or both true".

See also

• Boolean function
• If and only if
• Logical equivalence
• Logical biconditional
• Propositional calculus

External links

• Media related to Logical equality at Wikimedia Commons
• Mathworld, XNOR