relations

In mathematics we come across many relations like number m is less than number n, line p is perpendicular to line Q, set R is subset of set S. In all these, we notice that a relation involves pairs of objects.  In this lesson we will learn how to link pairs of objects from two sets and then introduce relations between the two objects in the pair.

Ordered Pair: An ordered pair consists of two objects or elements in a given fixed order. For example, if P and Q are two sets then by an ordered pair of elements we mean a pair (a,b) in that order, where a ∈ P, b ∈ Q

Equality of ordered pairs: Two ordered pair (a,b) and (c,d) are equal if  a = c and b = d

Cartesian product of sets 

Let A and B be any two non-empty sets. The set of all ordered pairs (a,b) such that a ∈ A and b ∈ B is called the cartesian products of the sets P and Q and is denoted by P × Q

Thus, P × Q = {(a,b) : a ∈ A and b ∈ B}

For example, if P = {2, 4, 6} and Q = {1, 2}, then

P × Q = {3, 5, 7} × {1, 2} = ((3, 1), (3, 2), (5, 1), (5, 2), (7, 1), (7, 2)}

Q × P = {1, 2} × {3, 5, 7}  =  ((1, 3), (1, 5), (1, 7), (2, 3), (2, 5), (2, 7)}

• Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal
• If there are m elements in P and n elements in Q, then there will be mn elements in P × Q.
• If P and Q are two non-empty sets and either P or Q is an infinite set, then so is P × Q

Example 1: If (x + 1, y − 3) = (4, 1), find the values of x and y. 

Solution: x + 1 = 4 therefore, x = 4 − 1 = 3

y − 3 = 1, therefore, y = 1 + 3 = 4

Example 2: If P = {1, 2, 3}, Q = {3, 4} and R = {1, 3, 5} find P × (Q ∪ R) 

Solution: Q ∪ R = {1, 3, 4, 5}

Therefore, P × (Q ∪ R) = {1, 2, 3} × {1, 3, 4, 5} = {(1, 1),(1, 3), (1, 4), (1, 5), (2,1), (2, 3), (2, 4), (2, 5), (3, 1), (3, 3), (3, 4), (3, 5)}

Relations

Consider two sets P and Q where P ={4, 9, 25} and Q = {+2, -2, +3, -3, +5, -5}

We can obtain a subset of P × Q  by introducing a relation R between the first element x and the second element y of each ordered pair (x,y) as

R = {(x,y):x is the square of the number y, x ∈ P and y ∈ Q}

A visual representation of this relation R (called an arrow diagram) is shown below:

In set builder form, R = {(x,y):x is the square of the number y, x ∈ P and y ∈ Q}

In roster form, R = {(4,+2), (4,-2), (9, +3), (9, -3), (25, +5), (25, -5)}

• Domain: the set of all first elements of the ordered pairs in relation R from a set P to set Q is called the domain of the relation R. In this example, the domain is {4, 9, 25}.
• Range and Codomain: The set of all second elements in a relation R from a set P to set Q is called the range of the relation R. The whole set Q is called the codomain of the relation R. Note that the range is a subset of the codomain. In the above example, range and codomain are the same and equal to {+2, -2, +3, -3, +5, -5}

Note: The total number of relations that can be defined from a set P to a set Q is the number of possible subsets of P × Q. If n(P) = r and n(Q) = s, then n(P × Q) = rs and the total number of relations is 2^rs

Example 3: Let P = {1, 2} and Q = {3, 4}. Find the number of relations from P to Q.

Solution: We have, P × Q = {(1,3), (1,4), (2,3), (2,4)}

Since n(P × Q) = 4, the number of subsets of P × Q is 2⁴, therefore, the number of relations will be 2⁴ = 16