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Question
Let A be the set of first five natural numbers and let R be a relation on A defined as follows:
(x, y) ∈ R ⇔ x ≤ y
Express R and R−1 as sets of ordered pairs. Determine also (i) the domain of R−1 (ii) the range of R.
Solution
Given:
A is the set of the first five natural numbers.
∴ A = {1, 2, 3, 4, 5}
The relation is defined as:
(x, y) ∈ R ⇔ x ≤ y
Now,
R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 4), (2, 5), (3, 3), (3, 4), (3, 5), (4, 4), (4, 5), (5, 5)}
R-1 = {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (2, 2), (3, 2), (4, 2), (5, 2), (3, 3), (4, 3), (5, 3), (4, 4), (5, 4), (5, 5)}
(i) Domain of R-1 = {1, 2, 3, 4, 5}
(ii) Range of R = {1, 2, 3, 4, 5}
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