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.