Let Z be the set of all integers and let R be a relation on Z defined by a R b⇔(a−b) is divisible by 3. Then, R is?

Let R={(a,b):a,b∈Z and (a−b) is divisible by 3}.Show that R is an equivalence relation on Z.Given:R={(a,b):a,b∈Z and (a−b) is divisible by 3}.R=(a,b)(a−b) is divisible by 3Reflexive(a,a)⇒(a−a) is divisible by 3Symmetric(a,b)⇔(b,a)(a−b) is divisible by 3(b−a) is divisible by 3Transitive(a,b),(b,c)⇔(a,c)(a−b) is divisible by 3(b−c) is divisible by 3(a−c) is divisible by 3R is an equivalent relation on Z

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Let Z be the set of all integers and let R be a relation on Z

Question

Let Z be the set of all integers and let R be a relation on Z defined by $a$ R $b \Leftrightarrow (a - b)$ is divisible by $3$ . Then, R is?

Reflexive and symmetric but not transitive

Reflexive and transitive but not symmetric

Symmetric and transitive but not reflexive

An equivalence relation

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Let Z be the set of all integers and let R be a relation on Z defined by

a

b \Leftrightarrow (a - b)

is divisible by

3

. Then, R is?

7 mins ago

Discuss this question LIVE

7 mins ago

Text solutionVerified

Let R={(a,b):a,b∈Z and (a−b) is divisible by 3}.

Show that R is an equivalence relation on Z.

Given:

R={(a,b):a,b∈Z and (a−b) is divisible by 3}.

R = (a, b)

(a - b)

is divisible by

3

Reflexive

(a, a) \Rightarrow (a - a)

is divisible by

3

Symmetric

(a, b) \Leftrightarrow (b, a)

(a - b)

is divisible by

3

(b - a)

is divisible by

3

Transitive

(a, b), (b, c) \Leftrightarrow (a, c)

(a - b)

is divisible by

3

(b - c)

is divisible by

3

(a - c)

is divisible by

3

R is an equivalent relation on Z

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Question Text	Let Z be the set of all integers and let R be a relation on Z defined by $a$ R $b \Leftrightarrow (a - b)$ is divisible by $3$ . Then, R is?
Answer Type	Text solution:1
Upvotes	150

Let Z be the set of all integers and let R be a relation on Z defined by a R b⇔(a−b) is divisible by 3. Then, R is?

Text solutionVerified

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Let Z be the set of all integers and let R be a relation on Z defined by $a$ R $b \Leftrightarrow (a - b)$ is divisible by $3$ . Then, R is?