relations and functions

relation

a mapping from elements of one set (the
domain) to elements of another (the codomain).

Arrow representation

where you have written out all the elements of both sets and connect them using arrows

A ordered pairs that are a subset of
𝐴 × 𝐵

{(3.-9),(0,-1),(2,16),(3,34),(4,40),(2,99)}

Tabular form

where there are 2 columns with both each set, values can be repeated in each column to show that they have relations with multiple values

Set builder notation

{(a,b)|b \= (+-)a^2, a ∈ R}

Graphically

if the values are numerical you can have one set on the x axis and one on the y

ways to represent a binary relation

Arrow representation,A ordered pairs that are a subset of
𝐴 × 𝐵, Tabular form, Set builder notation, Graphically

If a relation R is a subset of AxB and a relation S is a subset of BxC then

then R and S are composite relations

inverse relations

converse/opposite relation. essentially the arrows go a different direction

identity relation

relates 2 sets that are the same and related them normally {(x,x)|x ∈ A}, composition of a function and its inverse

A function 𝑓 from a set 𝐴 to a set B

an assignment of exactly
one element of b ∈ 𝐵 to each element of a ∈ 𝐴, f(a) \= b

single-valued

must map each input to at most one image b ∈ 𝐵

total

must map each possible input to at least one image b ∈ 𝐵

if a relation is single-values and total

each value 𝑎 ∈ 𝐴 must be mapped to exactly one value 𝑏 ∈
𝐵

If 𝑓: 𝐴→B what is the domain and codomain

𝐴 is the domain of 𝑓 and 𝐵 is the
codomain of f

If 𝑓(𝑎) \= 𝑏 what is the image and pre-image

the element 𝑏 is the image of element 𝑎,
and 𝑎 is a pre-image of 𝑏

range of 𝑓: 𝐴→𝐵

set of all images of elements of 𝐴

If A has size m and B has size n how many functions are there from A to B

n^m

an endorelation on a set A

is a relation from A to A

an endofunction on a set A

is a function from A to A

𝑓 is one-to-one(injective) iff

it does
not map two distinct elements of 𝐴 onto the same
element of B

f is surjective(onto) iff

for all elements of B there is an element in A with f(a) \= b
its range is its entire codomain

bijection or one-to-one correspondence

iff it is one-to-one and onto

(f o g)(x)

f(g(a))