Let A be the set of first five natural 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.

Question

Philoid · Accepted Answer

A is set of first five natural numbers.

Therefore, A= {1, 2, 3, 4, 5}

Given, (x, y) R x ≤ y

1 is less than 2, 3, 4 and 5.

2 is less than 3, 4 and 5.

3 is less than 4 and 5.

4 is less than 5.

5 is not less than any number A

∴ R = {(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)}

An inverse relation is the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original relation. If the graph of a function contains a point (a, b), then the graph of the inverse relation of this function contains the point (b, a).

∴ R^-1 = {(2, 1), (3, 1), (4, 1), (5, 1), (3, 2), (4, 2), (5, 2), (4, 3), (5, 3), (5, 4)}

⇒ R^-1 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3), (5, 4)}

i. Domain of R^‑1 = {2, 3, 4, 5}

ii. Range of R = {2, 3, 4, 5}

NOTE: You can see that Domain of R^‑1 is same as Range of R. Similarly, Domain of R is same as Range of R^‑1

Let A be the set of first five natural and let R be a relation on A defined as follows: (x, y) R x ≤ yExpress R and R-1 as sets of ordered pairs. Determine alsoi. the domain of R‑1ii. The Range of R.

Let A be the set of first five natural 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.