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> Functions
> Definitions
## DefinitionsThe definitions of functions and related terms.## Sibling topics:## Contents:- Definition of a function
- Definition of bijective
- Definition of composition
- Definition of image set
- Definition of injective
- Definition of strictly increasing
- Definition of surjective
Definition of a function
A function from `A` to `B` is set of ordered pairs `S sube A xx B` such that:
- `(AA u in A)(EE v in B)((u,v) in S)`, and
- `(AA u in A)(AA v,w in B)((u,v) in S and (u,w) in S => v=w)`
The set `A` is called the function's domain and the set `B` is called the function's
codomain. The above definition basically says that a) for every element `u` in `A`, `f(u) in B`,
and b) if `f(u)=v and f(u)=w`, then `v=w` (ie, each element of the function's domain maps to exactly one
element of the function's codomain). The word "range" is sometimes used to refer to a function's codomain,
and sometimes to refer to its image set.
Definition of image set
The image set of a function is the subset of the function's codomain
onto which elements of its domain are mapped.
The image set of a function `f:A->B`, is defined as `{f(t):t in A}`
and may be written as `Im(f) or f(A)`.
Definition of surjective
A function is surjective if the function's image set equals its codomain.
A function `f:A->B` is surjective (or " onto") if
`(AA v in B)(EE u in A)(f(u)=v)
More concisely, `f` is surjective if `Im(f)=B`. This essentially means that every element of the function's
codomain is mapped to by some element of its domain.
Definition of injective
A function is injective (or "one-to-one") if no element of its
codomain is mapped to by more than one element of its domain.
A function `f:A->B` is injective if `(AA u,w in A)(f(u)=f(w) => u=w)`
This means that no element of the function's codomain is mapped to by more than one
element of its domain. In order for a function to be invertible, it
must be one-to-one.
Definition of composition
Given `f:A->B` and `g:B->C`, the composition of `f` and `g` (written `g@f` and read "`g`
composition `f`" or "`g` of `f`") is the subset of `A xx C` equal to
`{(u,w) in A xx C : g(f(u))=w}`
In other words, `(u,w) in g@f` means `w=g(f(u))`, and `(g@f)(x) = g(f(x))` for all `x`.
Definition of strictly increasing
For a function `f` with domain `RR`, `f` is strictly increasing if:
`(AA x,y in RR)(x<y => f(x)<f(y))` |