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

Terminology

• An injective function is sometimes called an embedding
• Just as surjections are called 'onto' an injection may be called 'into'
  • This is French, from "throwing into" referring to the domain, not elements themselves (as any function takes an element into the codomain, it need not be one-to-one)
  • I do not like using the word into but do like onto - I say:

    "But $f$ maps $A$ onto $B$ so...."

    "But $f$ is an injection so...."

    "As $f$ is a bijection..."

  • I see into used rarely to mean injection.

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TODO: Investigate page 6 in Probability and Stochastics, it uses into and see if injectivity matters

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Properties

• The cardinality of the inverse of an element $$y\in Y$$ may be no more than 1
  • Note this means 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, writing $f^{-1}(y)=x$)

See also

• Bijection
• Surjection
• Function