Relations and Functions · Class 12 · CBSE · Science Mathematics
On Z, define a relation R by aRb if and only if a − b is even. Prove that R is an equivalence relation and find its equivalence classes.
Practice question from Relations and Functions (Class 12 · CBSE · Science Mathematics), with the correct answer and a step-by-step solution.
medium · Subjective · Q-85750D
On Z, define a relation R by aRb if and only if a − b is even. Prove that R is an equivalence relation and find its equivalence classes.
Answer: For every a ∈ Z, a − a = 0 is even, so R is reflexive; if a − b is even, then b − a is also even, so R is symmetric. If a − b and b − c are even, then a − c = (a − b) + (b − c) is even, so R is transitive. Hence R is an equivalence relation, with two equivalence classes: the even integers [0] and the odd integers [1].
Solution
The difference of two integers is even exactly when they have the same parity. This directly establishes reflexivity, symmetry, and transitivity, and separates Z into the even and odd equivalence classes.
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