Types of functions

Types of mappings

• a relation

• a function

  • every function is a relation

  • a function is a relation that is functional and serial

  • functional relation: left-unique (wrt to the ordered pairs)

  • serial relation: right-serial (wrt to the ordered pairs)

• injection: one-to-one mapping

• surjection: onto mapping

• bijective: one-one and onto mapping

• identity function

• inverse function

  • inverse of a mapping is not necessarily a function (inverse relation)

• total function: usual definition of a function

• partial function allows for unassociated elements in the domain

ajections

Mappings

• injection: one-to-one mapping

• surjection: onto mapping

• bijective: one-to-one and onto mapping

Injection: the main propery is that the mapping is one-to-one. Injection is a mapping f: A -> B associates a unique a ∈ A to a unique b in B:

• ∀a ∈ A. ∃!b ∈ B. f(a)=b

• ∀a,a' ∈ A. f(a) = f(a') <-> a = a'

Normally, a function can associate an element in a codomain to more then one element in a domain; that is, the same element b may be at the head of more then one arrow from A to B, f(a)=b ∧ f(a')=b ∧ a ≠ a'.

Surjection is a function with two properties, one "bad" and one "good". The good property ensures that the entire codomain is involved in the mapping - all elements in B are associated. The bad property is that some elements may be associated more then once.

Bijection takes an injection and the good property of surjection, so it is a mapping that is both one-to-one and onto (the entire codomain is associated).

Functional relations

Functions define many interesting relations between sets. Isomorphism is the most important relation:

Sets D and C are isomorphic if there exist a pair of functions f:D->C and g:C->D such that g . f is the identity function for D and f . g is the identity function for C. The maps (functions) f and g are called isomorphisms.

D = {...} f:D->C I(D) = g . f C = {...} g:C->D I(C) = f . g

D=[1,2,3,4], f:D->C, f(d)=c = f(d)=d*2 : f([1,2,3,4]) => [2,4,6,8] C=[2,4,6,8], g:C->D, g(c)=d = g(c)=c/2 : g([2,4,6,8]) => [1,2,3,4]

I(D) = I(d)=d : [1,2,3,4] => [1,2,3,4] I(C) = I(c)=c : [2,4,6,8] => [2,4,6,8]

I(D) = g.f ? g(f(d)) : g(f([1,2,3,4])) => g([2,4,6,8]) => [1,2,3,4] I(C) = f.g ? f(g(c)) : f(g([2,4,6,8])) => f([2,4,6,8]) => [2,4,6,8]

...thus, f and g are isomorphisms and the sets D and C are isomorphic.

A function is an isomorphism if and only if it is bijective i.e. both, one-one and onto.

The inverse f^-1 of isomorphism f is also an isomorphism, as f^-1 . f and f . f^-1 are both identities.