Let a set A = A₁ ∪ A₂ ∪ ..... ∪ Aₖ, where Aᵢ ∩ Aⱼ = φ for i ≠ j, 1 ≤ j, j ≤ k. Define the relation R from A to A by R = (x, y) : y ∈ Aᵢ if and only if x ∈ Aᵢ, 1 ≤ i ≤ k. Then, R is :

Solution

Maths sir · 4 steps

Read R as "x and y sit in the same piece", and all three properties fall out at once.

1. Decode the rule. The pieces A₁, A₂, …, Aₖ do not overlap and together cover A, so every element lies in exactly one piece. The condition "y ∈ Aᵢ if and only if x ∈ Aᵢ" then means x and y lie in the same piece.
2. Reflexive and symmetric. Every x is in the same piece as itself, so (x, x) ∈ R. If x and y share a piece, then y and x share it too, so (x, y) ∈ R gives (y, x) ∈ R.
3. Transitive. If x, y ∈ Aᵢ and y, z ∈ Aⱼ, then y lies in both Aᵢ and Aⱼ. Distinct pieces are disjoint, so i = j, and x, z ∈ Aᵢ gives (x, z) ∈ R.
4. So. R is reflexive, symmetric and transitive, so it is an equivalence relation, option (d).

The idea to keep. Split a set into non-overlapping pieces and relate elements in the same piece, and you always get an equivalence relation. Each piece is one equivalence class.

Answer from the NTA final answer key. Solution by PrepWiser.