Functions as a Set

If P and Q are two non-empty sets, then a function f from P to Q is a subset of P x Q, with two important restrictions

1. ∀ a ∈ P, (a, b) ∈ f for some b ∈ Q
2. If (a, b) ∈ f and (a, c) ∈ f then b = c.

Note1: There may be some elements of the Q which are not related to any element of set P.

2. Every element of P must be related with at least one element of Q.

Example1: If a set A has n elements, how many functions are there from A to A?

Solution: If a set A has n elements, then there are nⁿ functions from A to A.

Representation of a Function

The two sets P and Q are represented by two circles. The function f: P → Q is represented by a collection of arrows joining the points which represent the elements of P and corresponds elements of Q

Example1:

1. Let  X = {a, b, c}  and Y = {x, y, z}   and f: X → Y such that  
2.      f= {(a, x), (b, z), (c, x)}  

Then f can be represented diagrammatically as follows

Example2: Let X = {x, y, z, k} and Y = {1, 2, 3, 4}. Determine which of the following functions. Give reasons if it is not. Find range if it is a function.

1. f = {(x, 1), (y, 2), (z, 3), (k, 4)
2. g = {(x, 1), (y, 1), (k, 4)
3. h = {(x, 1), (x, 2), (x, 3), (x, 4)
4. l = {(x, 1), (y, 1), (z, 1), (k, 1)}
5. d = {(x, 1), (y, 2), (y, 3), (z, 4), (z, 4)}.

Solution:

1. It is a function. Range (f) = {1, 2, 3, 4}
2. It is not a function because every element of X does not relate with some element of Y i.e., Z is not related with any element of Y.
3. h is not a function because h (x) = {1, 2, 3, 4} i.e., element x has more than one image in set Y.
4. d is not a function because d (y) = {2, 3} i.e., element y has more than image in set Y.