Let A be the set of first five natural numbers and let R be a relation on A defined as follows:

Question:

Let A be the set of first five natural numbers and let R be a relation on A defined as follows:

(xy) ∈ 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:

(xy) ∈ 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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