If R is the empty relation on a non-empty set A, what is R∘R?
Answer and explanation
Correct answer: The empty relation
A pair (x,z) belongs to R∘R only if there exists some y such that (x,y) belongs to R and (y,z) also belongs to R. Since R contains no ordered pairs at all, no such intermediate y can exist. Consequently the composition is empty. The non-emptiness of A does not create pairs in an empty relation; it only ensures that the relation is not reflexive.
Frequently asked questions
What is the correct answer to this question?
The empty relation
Why is this the correct answer?
A pair (x,z) belongs to R∘R only if there exists some y such that (x,y) belongs to R and (y,z) also belongs to R. Since R contains no ordered pairs at all, no such intermediate y can exist. Consequently the composition is empty. The non-emptiness of A does not create pairs in an empty relation; it only ensures that the relation is not reflexive.
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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