Definition·D016
Bijection
A perfect matching: injective and surjective at once.
A function
is a bijection if it is injective and surjective onto
:
In words
f from A to B is a bijection exactly when f is injective and f is surjective onto B: every element of B is the value at exactly one element of A.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
A bijection is a complete, lossless matching between
and
: each element of either set corresponds to exactly one of the other. It is the precise sense in which two sets are "the same size", which is how equinumerosity and, for finite sets, cardinality are defined. Every bijection can be undone: its inverse is again a bijection (L03).
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