Logical equality: Difference between revisions

[[File:XNOR ANSI Labelled.svg|thumb|118px|XNOR Logiclogic Gategate Symbolsymbol]]

:<math>\begin{matrixalign}

x &\leftrightarrow y & \quad & \quadx & x \Leftrightarrow y & \quad & Exy \\

x &\ \mboxmathrm{~EQ~} \ y & \quad & \quadx & x = y

Some logicians, however, draw a firm distinction between a ''functional form'', like those in the lefthandleft 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 righthandright 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 (mathematics)|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:

: <math>\begin{matrixalign}

x &+ y & \quad & \quadx & x \not\equiv y & \quad & Jxy \\

x &\ \mboxmathrm{~XOR~} \ y & \quad & \quadx & x \ne y

This explains why "EQ" is often called "[[XNOR gate|XNOR]]" in the [[combinational logic]] of circuit engineers, since it is the ''Negationnegation'' of the ''[[XOR]]'' operation; '''NXOR''' is a less commonly used alternative.<ref>{{citation|title=Using Java 2|first1=Brian|last1=Keeton|first2=Chuck|last2=Cavaness|first3=Geoff|last3=Friesen|publisher=Que Publishing|year=2001|isbn=9780789724687|page=112|url=https://books.google.com/books?id=yhFxiVyd1MgC&pg=PA112}}.</ref> 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".