If R is transitive, so are PR and IR (prove

It seems like you forgot to specify the question or the properties of the relations P, I, and R. However, I can provide a general proof for the transitivity of the composition of two relations, assuming that R is transitive.

Let's denote the relations as follows:

• P: a relation from set A to set B
• R: a transitive relation from set B to set C
• I: a relation from set C to set D

We want to prove that the relations PR (from A to C through B) and IR (from B to D through C) are also transitive.

1. PR is transitive:

Assume (a, c) ∈ PR and (c, e) ∈ PR for some a, c, e ∈ A. By definition of the composition of relations, there exist b, d ∈ B such that (a, b) ∈ P, (b, c) ∈ R, (c, d) ∈ P, and (d, e) ∈ R. Since R is transitive, (b, d) ∈ R. Therefore, (a, d) ∈ P and (d, e) ∈ R, so (a, e) ∈ PR. Hence, PR is transitive.

2. IR is transitive:

Assume (b, d) ∈ IR and (d, f) ∈ IR for some b, d, f ∈ B. By definition of the composition of relations, there exist c, e ∈ C such that (b, c) ∈ I, (c, d) ∈ R, (d, e) ∈ I, and (e, f) ∈ R. Since R is transitive, (c, e) ∈ R. Therefore, (b, e) ∈ I and (e, f) ∈ R, so (b, f) ∈ IR. Hence, IR is transitive.

This proof assumes that the relations P and I are well-defined and that the sets A, B, C, and D are non-empty. If any of these conditions are not met, the proof may not hold.