Definition·D014
Injective function
A function sending distinct arguments to distinct values.
for a function
.
In words
f is injective exactly when for any x and x' with both in the domain A, if f takes the same value at x and at x' then x and x' are equal.
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
Also called one-to-one. Injectivity is what makes a function reversible on its range: each value is hit at most once, so it remembers where it came from (L03 exploits exactly this). The contrapositive reading is often the useful one:
implies
. The pigeonhole principle is the statement that no injection can squeeze a larger finite set into a smaller one.
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