Injection

An injective function is 1:1, but not nessasarally onto.

Definition

For a function $$f:X\rightarrow Y$$ 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 $$Y$$ . That is:

• $$\forall x_1,x_2\in X[f(x_1)=f(x_2)\implies x_1=x_2]$$

Or equivalently:

• $$\forall x_1,x_2\in X[x_1\ne x_2\implies f(x_1)=f(x_2)]$$ (the contrapositive of the above)

Notes

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)

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TODO: Find reference - should be easy!

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See also

• Bijection
• Surjection
• Function