Consider the following relations
$R_{1} (a, b)$ iff $(a + b)$ is even over the set of integers
$R_{2} (a, b)$ iff $(a + b)$ is odd over the set of integers
$R_{3} (a, b)$ iff $(a . b > 0)$ over the set of non-zero rational numbers.
$R_{4} (a, b)$ iff $| a - b | \leq 2$ over the set of natural numbers
Which of the following statements is correct?

R_{1}

and

R_{2}

are equivalence relations,

R_{3}

and

R_{4}

are not

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R_{1}

and

R_{3}

are equivalence relations,

R_{2}

and

R_{4}

are not

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R_{1}

and

R_{4}

are equivalence relations,

R_{2}

and

R_{3}

are not

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R_{1}, R_{2}, R_{3}

and

R_{4}

are all equivalence relations

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Solution

The correct option is B $R_{1}$ and $R_{3}$ are equivalence relations, $R_{2}$ and $R_{4}$ are not
(I) Relation $R_{1} (a, b)$ iff $(a + b)$ is even over the set of integers
(i) $a + a = 2 a$ which is even
So $(a, a)$ belongs to $R_{1}$
$∴ R_{1}$ is reflexive relation

(ii) If $(a + b)$ is even, then $(b + a)$ is also even
$∴ R_{1}$ is symmetric relation

(iii) If $(a + b)$ and $(b + c)$ are even then
$a + c = (a + b) + (b + c) - 2 b$
= even + even - even
= even
$∴ R_{1}$ is transitive relation
Since $R_{1}$ is reflexive, symmetric and transitive so $R_{1}$ is an equivalence relation

(II) $R_{2} (a, b)$ iff $(a + b)$ is odd over set of integers.
(i) $a + a = 2 a$ which is not odd
So $(a, a)$ doesn't belong to $R_{2}$
$∴ R_{2}$ is not reflexive relation
Since $R_{2}$ is not reflexive, it is not an equivalence relation.

(III) $R_{3} (a, b)$ iff $a . b > 0$ over set of non-zero relational numbers
(i) $a . a > 0$ for every non-zero rational number
$∴ R_{3}$ is reflexive relation.

(ii) If $a . b > 0$ then $b . a > 0$
$∴ R_{3}$ is symmetric relation

(iii) $a . b > 0$ and $b . c > 0 \Rightarrow$ All $a, b, c$ are positive or all $a, b, c$ are negative.
So $a . c > 0$
$∴ R_{3}$ is transitive relation.
So $R_{3}$ is an equivalence relation.

(IV) $R_{4} (a, b)$ iff $| a - b | \leq 2$ over the set of natural number
(i) $| a - a | \leq 2$
$0 \leq 2$
$∴ R_{4}$ is reflexive relation.

(ii) If $| a - b | \leq 2$ then also $| b - a | \leq 2$
$∴ R_{4}$ is symmetric relation

(iii) If $| a - b | \leq 2$ and $| b - c | \leq 2$
Ex. $| 3 - 5 | \leq 2$ and $| 5 - 7 | \leq 2$ but
$| 3 - 7 | / \leq 2$
$∴ R_{4}$ is not transitive
Since $R_{4}$ is reflexive and symmetric not transitive, so $R_{4}$ is not an equivalence relation.

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