If R and S are transitive relations on A, which implication proves that R∩S is transitive?
Answer and explanation
Correct answer: (a,b),(b,c)∈R∩S ⇒ (a,c)∈R∩S
Suppose (a,b) and (b,c) are in R∩S. Then both pairs are in R and in S. Since R is transitive, (a,c) is in R; since S is transitive, (a,c) is also in S. Therefore (a,c) belongs to their intersection, proving the implication in option A. Options B and C describe symmetry and reflexivity, which are not guaranteed, and D is far stronger than necessary.
Frequently asked questions
What is the correct answer to this question?
(a,b),(b,c)∈R∩S ⇒ (a,c)∈R∩S
Why is this the correct answer?
Suppose (a,b) and (b,c) are in R∩S. Then both pairs are in R and in S. Since R is transitive, (a,c) is in R; since S is transitive, (a,c) is also in S. Therefore (a,c) belongs to their intersection, proving the implication in option A. Options B and C describe symmetry and reflexivity, which are not guaranteed, and D is far stronger than necessary.
Which subject and chapter does this question cover?
This is a Class 12 Mathematics question. Chapter: Relations and Functions. Topic: Types of relations.
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