Final answer:
The question involves proving that a relation r is transitive if and only if the composition of r with itself is a subset of r, which is a concept in set theory. The proof relies on understanding the definitions of transitivity and composition of relations, ultimately concluding that the statement is true.
Step-by-step explanation:
The question pertains to the relationship between the properties of transitivity and composition in the context of a relation, often discussed in set theory or abstract algebra. The property of transitivity states that for a relation r on a set, if (a, b) is in r, and (b, c) is also in r, then (a, c) must be in r for r to be considered transitive. The operation r ◦ r represents the composition of r with itself, which involves creating pairs (a, c) where there exists some b such that (a, b) and (b, c) are both in r.
To prove that r is transitive if and only if r ◦ r ⊆ r, we need to establish two implications: If r is transitive, then r ◦ r must be a subset of r; and conversely, if r ◦ r is a subset of r, then r must be transitive. The first implication is straightforward - by the definition of transitivity, any pair derived from the composition r ◦ r will necessarily be in r. The second implication follows similarly since any pair in r ◦ r observes the transitive property and, by assumption, is also in r. Therefore, the statement is True.