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

$$f:X\rightarrow Y$$ every element of $$X$$ is mapped to an element of $$Y$$ and no two distinct things in $$X$$ are mapped to the same thing in $$Y$$ .

For this reason injectivity is often stated as

$$\forall x_1,x_2\in X:f(x_1)=f(x_2)\implies x_1=x_2$$

The cardinality of the inverse of an element

$$y\in Y$$ 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 $$f^{-1}(y)=\{x\}$$ as the value it contains)