Question

A relation R is defined on the set of integers as aRb if and only if a+b is even and R is termed as ______

(a) an equivalence relation with one equivalence class

(b) an equivalence relation with two equivalence classes

(c) an equivalence relation

(d) an equivalence relation with three equivalence classes

The question was asked in an interview for job.

I want to ask this question from Closure on Relations topic in division Relations of Discrete Mathematics

Answer 1

Right option is (b) an equivalence relation with two equivalence classes

Best explanation: R is reflexive as (a+b) is even for any integer; R is symmetric as if (a+b) is even (b+a) is also even; R is transitive as if ((a+b)+c) is even, then (a+(b+c)) is also even.

So, R is an equivalence relation. For set of natural numbers, sum of even numbers always give even, sum of odd numbers always give even and sum of any even and any odd number always give odd. So, must have two equivalence classes -> one for even and one for odd.

  {…, -4, -2, 0, 2, … } and {…, -3, -1, 1, 3, … }.