Functions and Types of Functions
Cartesian product of any two non-empty sets A and B denoted by A × B (read as A cross B) is a set containing all ordered pairs (x,y) such that x ∈ A and y ∈ B. We can write it as:
For example, if A = {a,b} and B = {2}, then
All the subsets of A × B are:
Any subset of the cartesian product A × B is called a binary relation (or simply a relation) from set A to set B. Basically, a binary relation relates an element of one set to an element of the other set. All the above four subsets are binary relations from set A to set B.
Domain of any relation R denoted by dom(R) is a set containing first element of each ordered pair in the relation. Dom(R₄) = {a,b}.
Range of a relation R denoted by ran(R) is a set containing second element of each ordered pair in the relation. Ran(R₄) = {2}.
In R₄, 2 is image of a and b but a and b are preimages of 2.
Function
A function f from set A to set B is a relation, rule or mapping which maps each element of set A to a unique element of set B such that domain of f = A and there is no repetition in the first element of ordered pairs in f.
Mathematically, it is written as f: A → B and is read as f is a function from A to B.
In the above example in which A = {a,b} and B = {2}
Consider R₄ = {(a,2), (b,2)}
Domain of R₄ = {a,b} = A
There is no repetition in the first element of ordered pairs in R₄.
Therefore, R₄ is a function.
The first element of each ordered pair in R₄ is called pre-image while its corresponding second element is called image of the first element. In (a, 2), a is the pre-image of 2 and 2 is the image of a.
Check Point:
R₁, R₂ and R₃ are not functions. Justify.
Domain of a Function
A set containing first element of each ordered pair in the function is called its domain.
Range of a Function
A set containing second element of each ordered pair in the function is called its range.