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:
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.
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, Epq, p ≡ q, or p == q) is as follows:
The form (x = y) is equivalent to the form (x ∧ y) ∨ (¬x ∧ ¬y).
For the operands x and y, the truth table of the logical equality operator is as follows:
y
T
F
x
T
T
F
F
F
T
Inequalityedit
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" 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:
This explains why "EQ" is often called "XNOR" in the combinational logic of circuit engineers, since it is the negation of the XOR operation; "NXOR" is a less commonly used alternative.[1] 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".
^Keeton, Brian; Cavaness, Chuck; Friesen, Geoff (2001), Using Java 2, Que Publishing, p. 112, ISBN9780789724687.
External linksedit
Media related to Logical equality at Wikimedia Commons
Mathworld, XNOR
December 18, 2023
logical, equality, corresponding, concept, combinational, logic, xnor, gate, logical, operator, that, corresponds, equality, boolean, algebra, logical, biconditional, propositional, calculus, gives, functional, value, true, both, functional, arguments, have, s. For the corresponding concept in combinational logic see XNOR gate 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 Logical equalityEQ XNORDefinitionx y displaystyle x y Truth table 1001 displaystyle 1001 Logic gateNormal formsDisjunctivex y x y displaystyle x cdot y overline x cdot overline y Conjunctive x y x y displaystyle overline x y cdot x overline y Zhegalkin polynomial1 x y displaystyle 1 oplus x oplus y Post s lattices0 preservingno1 preservingyesMonotonenoAffineyesvteIt 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 y x y E x y x E Q y x y displaystyle begin aligned x amp leftrightarrow y amp x amp Leftrightarrow y amp mathrm E xy x amp mathrm EQ y amp x amp 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 Contents 1 Definition 2 Alternative descriptions 3 Inequality 4 See also 5 References 6 External linksDefinition editLogical 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 Epq p q or p q is as follows nbsp The Venn diagram of A EQ B red part is true Logical equality p q p q0 0 10 1 01 0 01 1 1Alternative descriptions editThe form x y is equivalent to the form x y x y x y x y x y x y x y x y 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 nbsp For the operands x and y the truth table of the logical equality operator is as follows x y displaystyle x leftrightarrow y nbsp yT Fx T T FF F TInequality editIn 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 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 x y x y J x y x X O R y x y displaystyle begin aligned x amp y amp x amp not equiv y amp Jxy x amp mathrm XOR y amp x amp neq y end aligned nbsp This explains why EQ is often called XNOR in the combinational logic of circuit engineers since it is the negation of the XOR operation NXOR is a less commonly used alternative 1 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 edit nbsp Philosophy portal nbsp Psychology portalBoolean function If and only if Logical equivalence Logical biconditional Propositional calculusReferences edit Keeton Brian Cavaness Chuck Friesen Geoff 2001 Using Java 2 Que Publishing p 112 ISBN 9780789724687 External links edit nbsp Media related to Logical equality at Wikimedia Commons Mathworld XNOR Retrieved from https en wikipedia org w index php title Logical equality amp oldid 1168340790, wikipedia, wiki, book, books, library,
