Difference between revisions of "Injection"
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Revision as of 18:55, 12 February 2015
An injective function is 1:1, but not nessasarally onto.
For [math]f:X\rightarrow Y[/math] every element of [math]X[/math] is mapped to an element of [math]Y[/math] and no two distinct things in [math]X[/math] are mapped to the same thing in [math]Y[/math].
For this reason injectivity is often stated as [math]\forall x_1,x_2\in X:f(x_1)=f(x_2)\implies x_1=x_2[/math]
The cardinality of the inverse of an element [math]y\in Y[/math] may be no more than 1; that is it may be zero, in contrast to a bijection where the cardinality is always 1 (and thus we take the singleton set [math]f^{-1}(y)=\{x\}[/math] as the value it contains)
