Relations and Functions in Math

Study Notes

A function f from set A to set B is a relation that satisfies:
- Every element in A is related to exactly one element in B
- Domain of f = A
It is denoted by f: A → B or f(x)
f(a) is the image of a under f
Range of f: set of all images of elements in A

One-One Function (Injective): a function f: A → B is one-one if every element in B is the image of at most one element in A
Onto Function (Surjective): a function f: A → B is onto if every element in B is the image of at least one element in A
One-One Onto Function (Bijective): a function f: A → B is one-one onto if it is both one-one and onto
Constant Function: a function f: A → B is constant if every element in A has the same image in B
Identity Function: a function f: A → A is identity if every element in A is mapped to itself

A relation R is a subset of the Cartesian product of set A and set B, denoted by R(A, B) or simply R.
If (a, b) is an element of R, it means a is related to b by R.
The domain of R is the set of all elements in A that are related to some element in B.
The range of R is the set of all elements in B that are related to some element in A.

A function f is a relation that satisfies two conditions: every element in A is related to exactly one element in B, and the domain of f is equal to A.
It is denoted by f: A → B or f(x), where f(a) is the image of a under f.
The range of f is the set of all images of elements in A.

A function f: A → B is one-one if every element in B is the image of at most one element in A.
In other words, every element in B has a unique pre-image in A.

A function f: A → B is onto if every element in B is the image of at least one element in A.
In other words, every element in B has a pre-image in A.

A function f: A → B is one-one onto if it is both one-one and onto.
In other words, every element in B has a unique pre-image in A, and every element in A has a unique image in B.