The relation R = {(x, y) : x, y Z and x + y is even} is: [2025]

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Q.
The relation R = {(x, y) : x, y $\in$ Z and x + y is even} is: [2025]

1 reflexive and transitive but not symmetric.

2 reflexive and symmetric but not transitive.

3 symmetric and transitive but not reflexive.

4 an equivalence relation.

Ans.
(4)

R = {(x, y) : x, y $\in$ Z and x + y is even}

For reflexive : x + x = 2x is even

For symmetric : x + y = y + x is even

For transitive : x + y is even and y + z is even then z + x is also even.

So, the relation is an equivalence.

Solution

Q.
The relation R = {(x, y) : x, y $\in$ Z and x + y is even} is: [2025]

1 reflexive and transitive but not symmetric.

2 reflexive and symmetric but not transitive.

3 symmetric and transitive but not reflexive.

4 an equivalence relation.

Ans.
(4)

R = {(x, y) : x, y $\in$ Z and x + y is even}

For reflexive : x + x = 2x is even

For symmetric : x + y = y + x is even

For transitive : x + y is even and y + z is even then z + x is also even.

So, the relation is an equivalence.

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Solution

Q. The relation R = {(x, y) : x, y ∈ Z and x + y is even} is: [2025]

1 reflexive and transitive but not symmetric.

2 reflexive and symmetric but not transitive.

3 symmetric and transitive but not reflexive.

4 an equivalence relation.

Ans. (4) R = {(x, y) : x, y ∈ Z and x + y is even} For reflexive : x + x = 2x is even For symmetric : x + y = y + x is even For transitive : x + y is even and y + z is even then z + x is also even. So, the relation is an equivalence.

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Q.
The relation R = {(x, y) : x, y $\in$ Z and x + y is even} is: [2025]

Ans.
(4)

R = {(x, y) : x, y $\in$ Z and x + y is even}

For reflexive : x + x = 2x is even

For symmetric : x + y = y + x is even

For transitive : x + y is even and y + z is even then z + x is also even.

So, the relation is an equivalence.