Question

A relation $R$ is defined on the set of integers as $xRy$ iff $(x+y)$ is even. Which of the following statements is true?

• A. $R$ is not an equivalence relation
• B. $R$ is an equivalence relation having 1 equivalence class
• C. $R$ is an equivalence relation having 2 equivalence classes
• D. $R$ is an equivalence relation having 3 equivalence classes

Answer 1

The correct option is C $R$ is an equivalence relation having 2 equivalence classes

A relation $R$ is defined as $xRy$ iff $(x+y)$ is ever over set of integers.
$(x+y)$ is even iff
(i) both $x$ and $y$ are even
(ii) both $x$ and $y$ are odd
Therefore, relation $R$ is equivalence relation because relation is
(i) Reflexive
$x+x=2x=$ even
So $(x.x)$ belongs to $R$. So relation is reflexive

(ii) Symmetric
If $x+y=$ even then $y+x$ is also even So relation is symmetric.

(iii) Transitive
If $x+y=$ even and $y+z=$ even
Then $x+y+y+z=$ even + even
$\Rightarrow x+z+2y=$ even
$\Rightarrow x+z=$ even - $2y$
$\Rightarrow x+z=$ even
$\therefore$ Relation $R$ is transitive. So relation $R$ is an equivalence relation which divides the set of integer into two equivalence classes: One is of all even and other is of odd integer.