Functions

Discrete Math

Let A and B be Sets. A function from A to B is an assignment of a unique element of B to each element of A.

If f is a function from A to B, we say that a is the domain of f and B is the codomain of f. If f(a)=bf(a) = b we say that b is the image of a and a is a pre-image of b. The range of f is the set of all images of elements of A.

A function f is said to be one-to-one or injective, if f(x)≠f(y)f(x) ≠ f(y) whenever x≠yx ≠ y. A function that is not one-to-one is not invertible.

A function f is said to be onto or surjective, IFF for every element b∈Bb ∈ B there is an element a∈Aa ∈ A with f(a)=bf(a) = b.

The function f is a one-to-one correspondence or a bijection if it is both one-to-one and onto.

Function composition: (f∘g)(a)=f(g(a))(f ∘ g)(a) = f(g(a)) (f−1∘f)(a)=a(f^{-1} ∘ f)(a) = a

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