On the real numbers, define aRb if and only if a^3 = b^3. What type of relation is R?
Answer and explanation
Correct answer: Equivalence relation
For real numbers, the function x ↦ x^3 is one-to-one. Thus a^3 = b^3 implies a = b, so R is exactly the identity relation {(a,a): a∈R}. Equality is reflexive, because a^3=a^3; symmetric, because a^3=b^3 implies b^3=a^3; and transitive, because equal cubes pass through an intermediate value. Hence option A is correct, while the other statements deny properties that R has.
Frequently asked questions
What is the correct answer to this question?
Equivalence relation
Why is this the correct answer?
For real numbers, the function x ↦ x^3 is one-to-one. Thus a^3 = b^3 implies a = b, so R is exactly the identity relation {(a,a): a∈R}. Equality is reflexive, because a^3=a^3; symmetric, because a^3=b^3 implies b^3=a^3; and transitive, because equal cubes pass through an intermediate value. Hence option A is correct, while the other statements deny properties that R has.
Which subject and chapter does this question cover?
This is a Class 11 Mathematics question. Chapter: Relations and Functions. Topic: Relations.
