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In the set Z of all integers, which of the following relation

Question

In the set $Z$ of all integers, which of the following relation $R$ is an equivalence relation?

x R y :

x \leq y

x R y :

x - = y

x R y :

x - y

is an even integer

x R y :

x \equiv y

(mod 3)

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In the set

Z

of all integers, which of the following relation

R

is an equivalence relation?

5 mins ago

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Text solutionVerified

For option C, xRy:if x-y is an even integer .

Let a be any number in Z,

Then, a-a=0 is an even integer .

So, Every number is related to itself .

Thus the relation is reflexive .

Let a,b be two numbers in Z,

Also let

a - b = k

be an even integer (which implies aRb)

Then,

b - a = - k

is also an even integer (which implies bRa).

Thus, It is evident

a R b \leftrightarrow b R A

Thus the relation is also symmetric .

Let a,b,c be three numbers in Z,

Also let aRb and bRc,

then

a - b = 2 k_{1}

and

b - c = 2 k_{2}

for

k_{1}, k_{2}

belonging to Z

which gives

a - b + b - c = 2 k_{1} + 2 k_{2} \Rightarrow a - c = 2 (k_{1} + k_{2})

which implies

a R c

Thus relation is also transitive .

Thus we finally arrive at a conclusion that relation is Equivalence relation .

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Question Text	In the set $Z$ of all integers, which of the following relation $R$ is an equivalence relation?
Answer Type	Text solution:1
Upvotes	150

In the set Z of all integers, which of the following relation R is an equivalence relation?

Text solutionVerified

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In the set $Z$ of all integers, which of the following relation $R$ is an equivalence relation?