**A very important special type of relation is a function defined as below:**

**Function in Math**: If A and B are two sets the relation f:A\rightarrow B is called function if for every element of A there exist a unique element of B, and Dom f = A.

f:A\rightarrow B which is read: f is a function from A to B

If (x, y) in an element of f when regarded as a set of ordered pairs,

we write y = f (x). y is called the value of f for x or image of x under f.

In example 1 discussed above

i) r is a subset of C\times F

ii) Dom\;r\;=\;\left{c_1,c_2,c_3\right}=C

iii) First elements of no two related pairs of r are the same.

Therefore, r is a function from C to F.

In Example 2 relation topic check

i) r is a subset of A\times A;

ii) Dom r ≠ A

Therefore, the relation in this case is not a function.

In example 3 relation topic check

i) r is a subset of \mathfrak R

ii) Dom r =\mathfrak R

iii) Clearly first elements of no two ordered pairs of r can be equal. Therefore, in this case

r is a function

Into Function: If a function f:A\rightarrow B is such that Range\;f\;\subset\;B i.e., Range\;f\;\neq\;B, then f is said to be a function from A

into B. In fig.(1) f is clearly a function. But Range\;f\;\neq\;B

Therefore, f is a function from A into B.