Let n be a fixed positive integer. Define a relation R on Z as follows:(a,b)∈R⇔ n divides a−b.Show that R is an equivalence relation on Z.

Question

Filo · Accepted Answer

Let n be a fixed positive integer.Let the relation be R on Z is given by R=\left\{ \left( a,b ight) :n\quad 	ext{divides}\quad a-b ight\} .We observe the following properties of relation RReflexivity: for any a\in Za-a=0=0	imes n\Rightarrow n divides a-a\Rightarrow (a,a)\in RSo, R is relexive relation on ZSymmetry: Let a,b\in Z be such that(a,b)\in R\Rightarrow n divides a-b\Rightarrow a-b=n\lambda for some \lambda \in Z\Rightarrow b-a=n(-\lambda), where -\lambda \in Z\Rightarrow n divides b-a\Rightarrow (b,a)\in RThus (a,b)\in R\Rightarrow (b,a)\in R. So, R is a symmetric relation on Z.Transitivity. Let a,b,c\in Z be such that (a,b)\in R and (b,c)\in R. Then,(a,b)\in R\Rightarrow n divides a-b\Rightarrow a-b=n\lambda for some \lambda \in Zand (b,c)\in R\Rightarrow n divides b-c\Rightarrow b-c=n\mu for some \mu \in Z	herefore a-b+b-c=n(\lambda +\mu)\Rightarrow a-c=n(\lambda +\mu), where \lambda +\mu \in Z\Rightarrow n divides a-c\Rightarrow (a,c)\in Rthus, (a,b)\in R and (b,c)\in R\Rightarrow (a,c)\in RSo, R is a transitive relation on ZHence, R is an equivalence relation on Z.

Question Text	Let $n$ be a fixed positive integer. Define a relation $R$ on $Z$ as follows: $(a, b) \in R \Leftrightarrow$ $n$ divides $a - b$ . Show that $R$ is an equivalence relation on $Z$ .
Answer Type	Text solution:1
Upvotes	150

Let $n$ be a fixed positive integer. Define a relation $R$ on $Z$ as follows:
$(a, b) \in R \Leftrightarrow$ $n$ divides $a - b$ .
Show that $R$ is an equivalence relation on $Z$ .

Text solutionVerified

Practice questions from similar books