Editing Logical connective (section)

==Applications==
Logical connectives are used in [[computer science]] and in [[set theory]].

===Computer science===
{{Main article|Logic gate}}
A truth-functional approach to logical operators is implemented as [[logic gate]]s in [[digital circuit]]s. Practically all digital circuits (the major exception is [[DRAM]]) are built up from [[logical NAND|NAND]], [[logical NOR|NOR]], [[negation|NOT]], and [[logic gate|transmission gate]]s; see more details in [[Truth function#Computer science|Truth function in computer science]]. Logical operators over [[bit array|bit vectors]] (corresponding to finite [[Boolean algebra (structure)|Boolean algebras]]) are [[bitwise operation]]s.

But not every usage of a logical connective in [[computer programming]] has a Boolean semantic. For example, [[lazy evaluation]] is sometimes implemented for {{math|''P'' ∧ ''Q''}} and {{math|''P'' ∨ ''Q''}}, so these connectives are not commutative if either or both of the expressions {{mvar|P}}, {{mvar|Q}} have [[side effect (computer science)|side effect]]s. Also, a [[conditional (programming)|conditional]], which in some sense corresponds to the [[material conditional]] connective, is essentially non-Boolean because for <code>if (P) then Q;</code>, the consequent&nbsp;Q is not executed if the [[antecedent (logic)|antecedent]]&nbsp;P is false (although a compound as a whole is successful ≈ "true" in such case). This is closer to intuitionist and [[constructive mathematics|constructivist]] views on the material conditional— rather than to classical logic's views.

===Set theory===
{{Main article|Set theory|Axiomatic set theory}}
Logical connectives are used to define the fundamental operations of [[set theory]],<ref>{{Cite book |last=Pinter |first=Charles C. |title=A book of set theory |date=2014 |publisher=Dover Publications, Inc |isbn=978-0-486-49708-2 |location=Mineola, New York |pages=26–29}}</ref> as follows:

{| class="wikitable" style="margin:1em auto; text-align:left;"
|+Set theory operations and connectives
|-
! Set operation
! Connective 
! Definition
|-
| [[Intersection (set theory)|Intersection]]
| [[Logical conjunction|Conjunction]]
| <math>A \cap B = \{x : x \in A \land x \in B \}</math><ref name=":0">{{Cite web |title=Set operations |url=https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-operations.html |access-date=2024-06-11 |website=www.siue.edu}}</ref><ref name=":1">{{Cite web |title=1.5 Logic and Sets |url=https://www.whitman.edu/mathematics/higher_math_online/section01.05.html |access-date=2024-06-11 |website=www.whitman.edu}}</ref><ref>{{Cite web |title=Theory Set |url=https://mirror.clarkson.edu/isabelle/dist/library/HOL/HOL/Set.html |access-date=2024-06-11 |website=mirror.clarkson.edu}}</ref>
|-
| [[Union (set theory)|Union]]
| [[Logical disjunction|Disjunction]]  
| <math>A \cup B = \{x : x \in A \lor x \in B \}</math><ref>{{Cite web |title=Set Inclusion and Relations |url=https://autry.sites.grinnell.edu/csc208/readings/set-inclusion.html |access-date=2024-06-11 |website=autry.sites.grinnell.edu}}</ref><ref name=":0" /><ref name=":1" />
|-
| [[Complement (set theory)|Complement]]
| [[Negation]] 
| <math>\overline{A} = \{x : x \notin A \}</math><ref>{{Cite web |title=Complement and Set Difference |url=https://web.mnstate.edu/peil/MDEV102/U1/S6/Complement3.htm |access-date=2024-06-11 |website=web.mnstate.edu}}</ref><ref name=":1" /><ref>{{Cite web |last=Cooper |first=A. |title=Set Operations and Subsets – Foundations of Mathematics |url=https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/ |access-date=2024-06-11 |language=en-US}}</ref>
|-
| [[Subset]]
| [[Material conditional|Implication]]
| <math>A \subseteq B \leftrightarrow (x \in A \rightarrow x \in B)</math><ref name=":2">{{Cite web |title=Basic concepts |url=https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-basics.html |access-date=2024-06-11 |website=www.siue.edu}}</ref><ref name=":1" /><ref>{{Cite web |last=Cooper |first=A. |title=Set Operations and Subsets – Foundations of Mathematics |url=https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/ |access-date=2024-06-11 |language=en-US}}</ref> 
|-
| [[Equality (mathematics)|Equality]]
| [[Logical biconditional|Biconditional]] 
| <math>A = B \leftrightarrow (\forall X)[A \in X \leftrightarrow B \in X]</math><ref name=":2" /><ref name=":1" /><ref>{{Cite web |last=Cooper |first=A. |title=Set Operations and Subsets – Foundations of Mathematics |url=https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/ |access-date=2024-06-11 |language=en-US}}</ref>
|}
This definition of set equality is equivalent to the [[axiom of extensionality]].