If R₂ = {(x, y) | x and y are integers and x² + y² = 64} is a relation. Then find R₂.

Given: R₂ = {(x, y) | x and y are integers and x² + y² = 64} is a relation

To find: R₂

Explanation: Given R₂ = {(x, y) | x and y are integers and x² + y² = 64}

This means set R₂ contains all whole numbers such that the relation between the elements of the set satisfy the condition x² + y² = 64, so

x² + y² = 64

⇒ y² = 64- x²

This is possible only when

64-x²≥0

⇒ 64≥ x²

⇒ x²≤64

Taking square root on both sides, we get

⇒ x≤±√64

⇒ x≤±8

i.e., -8≤x≤8

So, x ∈ [-8,8]

Now for corresponding values of x, y becomes

When x = ±8

So, two of the elements of R₂ = (-8, 0), (8, 0)

When x = ±7

Now as y is an integer, so x cannot take values, ±7

When x = ±6

Now as y is an integer, so x cannot take values, ±6

When x = ±5

Now as y is an integer, so x cannot take values, ±5

When x = ±4

Now as y is an integer, so x cannot take values, ±4

When x = ±3

Now as y is an integer, so x cannot take values, ±3

When x = ±2

Now as y is an integer, so x cannot take values, ±2

When x = ±1

Now as y is an integer, so x cannot take values, ±5

When x = 0

So, two of the element of R₂ = (0, 8), (0, -8)

Hence complete set is

R₂ = {(0, 8), (0, -8), (-8,0), (8, 0)}