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Check whether the relation R on R defined as R = {(a, b): a ≤ b3} is reflexive, symmetric or transitive.
Check whether the relation R on R defined as R = {(a, b): a ≤ b3} is reflexive, symmetric or transitive.
CBSE | Class 12 | Exercise 1.1 | Maths | RD Sharma | Relations
Check whether the relation R on R defined as R = {(a, b): a ≤ b3} is reflexive, symmetric or transitive.

Solution:

Given R = {(a, b): a ≤ b3}

It is observed that (1/2, 1/2) in R as 1/2 > (1/2)3 = 1/8

∴ R is not reflexive.

Now,

(1, 2) ∈ R (as 1 < 23 = 8)

But,

(2, 1) ∉ R (as 2 > 13 = 1)

∴ R is not symmetric.

We have (3, 3/2), (3/2, 6/5) in “R as” 3 < (3/2)3 and 3/2 < (6/5)3

But (3, 6/5) ∉ R as 3 > (6/5)3

∴ R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.

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