Definition·D037
Binary operation
A rule combining any two elements of a set into a third.
In words
A binary operation is a function from pairs of elements of a set to that same set.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Closure is built into the definition: the value
lands back in
, always. Addition and multiplication are binary operations on
; composition is a binary operation on the set of bijections of any fixed set. An operation by itself has no laws; asking for associativity, an identity and inverses is what defines a group.
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