Question:
A relation R is defined from the set of integers to the set of real numbers as follows : (x,y) = R if x^2 y^2 = 16 . Express R and R^-1 as sets of ordered pairs and hence find their ranges
Answer:
Given that:
(x, y) ∈ R ⇔ x2 + y2 = 16
⇔ y = ±√(16 - x2 )
when x = 0 ⇒ y = ±4
(0, 4) ∈ R and (0, -4) ∈ R
when x = ±4 ⇒ y = 0
(4, 0) ∈ R and (-4, 0) ∈ R
Now for other integral values of x, y is not an integer.
Hense R = {(0, 4), (0, -4), (4, 0), (-4, 0)}
Now R-1 = {(4, 0), (-4, 0), (0, 4), (0, -4)}
So Domain(R) = {0, -4, 4} = Range(R-1 )
and Range(R) = {4, -4, 0} = Domain(R-1 )
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