Editing Symmetry (section)

===In logic===

A [[binary relation|dyadic relation]] ''R'' = ''S'' × ''S'' is symmetric if for all elements ''a'', ''b'' in ''S'', whenever it is true that ''Rab'', it is also true that ''Rba''.<ref>Josiah Royce, Ignas K. Skrupskelis (2005) ''The Basic Writings of Josiah Royce: Logic, loyalty, and community (Google eBook)'' Fordham Univ Press, p. 790</ref> Thus, the relation "is the same age as" is symmetric, for if Paul is the same age as Mary, then Mary is the same age as Paul.

In propositional logic, symmetric binary [[logical connective]]s include ''[[logical conjunction|and]]'' (∧, or &), ''[[logical disjunction|or]]'' (∨, or |) and ''[[if and only if]]'' (↔), while the connective ''if'' (→) is not symmetric.<ref>{{Cite web|url=https://cs.uwaterloo.ca/~a23gao/cs245_f19/slides/lec02_prop_syntax_nosol.pdf|title=Propositional Logic: Introduction and Syntax|last=Gao|first=Alice|date=2019|website=University of Waterloo — School of Computer Science|access-date=2019-11-12}}</ref> Other symmetric logical connectives include ''[[logical nand|nand]]'' (not-and, or ⊼), ''[[xor]]'' (not-biconditional, or ⊻), and ''[[logical nor|nor]]'' (not-or, or ⊽).