Relations

A relation can be represented using set notation as:

R = {(a, b): a ∈ A, b ∈ B}

Example: From the table below, the relation is: {(-3, -6), (0.5, 1), (1, 2), (2, 4), (5, 10), (6, 12)}.

This relation satisfies the equation:

y = 2x

So, the relation R can be written symbolically as:

R = {(x, y): y = 2x}

Question:

Which of the following ordered pairs belong to the relation

{(x, y): y > x}?

(1, 2), (2, 1), (-3, 4), (-3, -5), (2, 2), (-8, 0), (-8, -3)

Solution:

Pairs where y > x are:

• (1, 2)
• (-3, 4)
• (-8, 0)

Example 1

Consider numbers 1, 2, 3, 4, 5, and 6.

Let:

• Set A = {1, 3, 5}
• Set B = {2, 4, 6}

The relation "is greater than" can be represented as an arrow diagram, showing which elements in A are greater than elements in B.

Relation as ordered pairs

A relation can also be described as a set of ordered pairs (a, b), where:

• a ∈ Set A
• b ∈ Set B
• a is related to b

Example 2

Let:

• Set A = {2, 3, 5, 6}
• Set B = {6, 7, 10}

Define the relation "is a factor of"

Then the relation will include pairs like:

• (2, 6)
• (2, 10)
• (3, 6)
• (5, 10)
• (6, 6)

These pairs form the set: R = {(2, 6), (2, 10), (3, 6), (5, 10), (6, 6)}

Example 3: Draw an arrow diagram

Question:

Let set A = {1, 2, 3, 4}

Draw an arrow diagram to show the relation connecting each element in set A with its square.

Solution: Let set B = {1, 4, 9, 16}

Then:

• 1 → 1
• 2 → 4
• 3 → 9
• 4 → 16

Ordered pairs: {(1, 1), (2, 4), (3, 9), (4, 16)}