Question

Let x be a nonvoid set.If ρ1 and ρ2 be the transitive relations of x, then-(° denotes the compositions of relations )

• A. ρ1°ρ2 is trasitive relation
• B. ρ1°ρ2 is not transitive relation
• C. ρ1°ρ2 is equivalence relation
• D. ρ1°ρ2 is not any relation on X

Answer 1

Answer: (B)

ρ1°ρ2 is not transitive relation

Answer 2

the composition of two transitive relations may not necessarily result in a transitive relation.

Transitivity means that if there is a relation R such that (a, b) ∈ R and (b, c) ∈ R, then (a, c) must also be in R.

Let's consider an example where this property does not hold for the composition of two transitive relations ρ1 and ρ2.

Suppose we have the set x = {1, 2, 3}, and let's define the relations ρ1 and ρ2 as follows:

ρ1 = {(1, 2), (2, 3)} ρ2 = {(2, 3), (3, 1)}

Both ρ1 and ρ2 are transitive relations because they satisfy the transitivity property on their own:

• For ρ1: (1, 2) ∈ ρ1 and (2, 3) ∈ ρ1, so (1, 3) ∈ ρ1 as well.
• For ρ2: (2, 3) ∈ ρ2 and (3, 1) ∈ ρ2, so (2, 1) ∈ ρ2 as well.

However, let's consider the composition ρ1°ρ2:

ρ1°ρ2 = {(1, 1), (1, 3), (2, 1), (2, 3), (3, 1), (3, 3)}

Note: that (1, 3) ∈ ρ1°ρ2 and (3, 1) ∈ ρ1°ρ2, but (1, 1) ∉ ρ1°ρ2. This breaks the transitivity property, as the element (1, 1) should be in the relation if (1, 3) and (3, 1) are in the relation.

This example demonstrates that the composition of the transitive relations ρ1 and ρ2, namely ρ1°ρ2, is not transitive. Thus, the answer is justified: ρ1°ρ2 is not a transitive relation.

The correct option is(B) ρ1°ρ2 is not transitive relation