If R is symmetric, which statement about R⁻¹ is correct?
Answer and explanation
Correct answer: R⁻¹ = R
The inverse relation R⁻¹ is formed by reversing every ordered pair: (a, b) ∈ R⁻¹ exactly when (b, a) ∈ R. If R is symmetric, the presence of (b, a) in R guarantees the presence of (a, b) in R, so reversing all pairs produces no new relation; every pair of R⁻¹ is already in R. The same reasoning in reverse gives R ⊆ R⁻¹, hence R⁻¹ = R. Symmetry does not imply emptiness, so options B and C are false, while D states the opposite. Therefore A is correct.
Frequently asked questions
What is the correct answer to this question?
R⁻¹ = R
Why is this the correct answer?
The inverse relation R⁻¹ is formed by reversing every ordered pair: (a, b) ∈ R⁻¹ exactly when (b, a) ∈ R. If R is symmetric, the presence of (b, a) in R guarantees the presence of (a, b) in R, so reversing all pairs produces no new relation; every pair of R⁻¹ is already in R. The same reasoning in reverse gives R ⊆ R⁻¹, hence R⁻¹ = R. Symmetry does not imply emptiness, so options B and C are false, while D states the opposite. Therefore A is correct.
Which subject and chapter does this question cover?
This is a Class 12 Mathematics question. Chapter: Relations and Functions. Topic: Symmetric relation.
