< Logical Equality(logic, wiki, imported, Proteus)
missing image!
- XNOR.jpg -
XNOR Logic Gate Symbol
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:
x leftrightarrow y & quad & quad & x Leftrightarrow y
x mbox{EQ} y & quad & quad & x = y
end{matrix}
Some logicians, however, draw a firm distinction between a
functional form, like those in the lefthand 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 righthand 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.
In
mathematics, the plus sign "+" almost invariably indicates an operation that satisfies the axioms assigned to addition in the type of
algebraic structure that is known as a
field. For boolean algebra, this means that the logical operation signified by "+" is not the same as the
inclusive disjunction signified by "∨" but is actually equivalent to the logical inequality operator signified by "≠", or what amounts to the same thing, the
exclusive disjunction signified by "XOR". 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:
x + y & quad & quad & x notequiv y
x mbox{XOR} y & quad & quad & x ne y
end{matrix}
This explains why "EQ" is often called "
XNOR" in the
combinational logic of circuit engineers, since it is the
Negation of the
XOR operation. 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".
Definition
Logical equality is an
operation on two
logical values, typically the values of two
propositions, 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, or
p ≡ q) is as follows:
|+ Logical Equality
style="background:paleturquoise"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p = q
| | T
|
| | F
|
| | F
|
| | T
|
Alternative descriptions
The form (
x =
y) is equivalent to the form (
x ∧
y) ∨ (¬
x ∧ ¬
y).
(x = y) = ˜(x + y) = (x ∧ y) ∨ (˜ x ∧ ˜ y)
For the operands
x and
y, the
truth table of the logical equality operator is as follows:
! colspan="2" rowspan="2" | x ↔ y
!! colspan="2" | y
! T !! F
! rowspan="2" | x !! T
| T
| F
|
! F
| F
| T
|
See also
Logical operators
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Related topics
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External links
Some content adapted from the Wikinfo article "Logical equality" under the GNU Free Documentation License.
XNOR-Gatter
Puerta lógica#Puerta equivalencia (XNOR)
XNOR-poort
XNOR kapısı
(last updated by Proteus, 6:43pm EDT - Sat, Apr 07 2007)
