## Math

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Math
> Functions
> Definitions
For sets `A` and `B`, "`f:A->B`" means `f` is a function from `A` to `B`. If `f` is a function, then
"`f(u)=v`" means `(u,v) in f`.

The set `A` is called the function's

The image set of a function `f:A->B`, is defined as

A function `f:A->B` is surjective (or "

A function `f:A->B` is injective if

### Definitions

The 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))`