Show that relation mathrm{R} in set mathrm{Z} of integers given by mathrm{R} ={(mathrm{a}, mathrm{b}

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1.Relation and Function

easy

Show that the relation $\mathrm{R}$ in the set $\mathrm{Z}$ of integers given by $\mathrm{R} =\{(\mathrm{a}, \mathrm{b}): 2$ divides $\mathrm{a}-\mathrm{b}\}$ is an equivalence relation.

Option A

Option B

Option C

Option D

Solution

$\mathrm{R}$ is reflexive, as $2$ divides $(\mathrm{a}-\mathrm{a})$ for all $\mathrm{a} \in \mathrm{Z}$. Further, if $(\mathrm{a}, \mathrm{b}) \in \mathrm{R} ,$ then $2$ divides $\mathrm{a}-\mathrm{b}$. Therefore, $2$ divides $\mathrm{b} – \mathrm{a}$. Hence, $(\mathrm{b}, \mathrm{a}) \in \mathrm{R}$, which shows that $\mathrm{R}$ is symmetric. Similarly, if $(\mathrm{a}, \mathrm{b}) \in \mathrm{R}$ and $(\mathrm{b}, \mathrm{c}) \in \mathrm{R} ,$ then $\mathrm{a}-\mathrm{b}$ and $\mathrm{b}-\mathrm{c}$ are divisible by $2$. Now, $\mathrm{a}-\mathrm{c}=(\mathrm{a}-\mathrm{b})+(\mathrm{b}-\mathrm{c})$ is even (Why?). So, $(\mathrm{a}-\mathrm{c})$ is divisible by $2 .$ This shows that $\mathrm{R}$ is transitive. Thus, $\mathrm{R}$ is an equivalence relation in $\mathrm{Z}$.

Standard 12

Mathematics

The relation $R$ defined in $N$ as $aRb \Leftrightarrow b$ is divisible by $a$ is

easy

Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) R (u, v)$ if and only if $x v=y u .$ Show that $R$ is an equivalence relation.

easy

Let $R$ be the relation in the set $N$ given by $R =\{(a,\, b)\,:\, a=b-2,\, b>6\} .$ Choose the correct answer.

easy

Determine whether each of the following relations are reflexive, symmetric and transitive :

Relation $\mathrm{R}$ in the set $\mathrm{A}=\{1,2,3, \ldots, 13,14\}$ defined as $\mathrm{R}=\{(x, y): 3 x-y=0\}$

medium

Given a non empty set $X$, consider $P ( X )$ which is the set of all subsets of $X$.

Define the relation $R$ in $P(X)$ as follows :

For subsets $A,\, B$ in $P(X),$ $ARB$ if and only if $A \subset B .$ Is $R$ an equivalence relation on $P ( X ) ?$ Justify your answer.

hard

Show that the relation $\mathrm{R}$ in the set $\mathrm{Z}$ of integers given by $\mathrm{R} =\{(\mathrm{a}, \mathrm{b}): 2$ divides $\mathrm{a}-\mathrm{b}\}$ is an equivalence relation.

Solution

Similar Questions

The relation $R$ defined in $N$ as $aRb \Leftrightarrow b$ is divisible by $a$ is

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Given a non empty set $X$, consider $P ( X )$ which is the set of all subsets of $X$. Define the relation $R$ in $P(X)$ as follows : For subsets $A,\, B$ in $P(X),$ $ARB$ if and only if $A \subset B .$ Is $R$ an equivalence relation on $P ( X ) ?$ Justify your answer.

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Determine whether each of the following relations are reflexive, symmetric and transitive :

Relation $\mathrm{R}$ in the set $\mathrm{A}=\{1,2,3, \ldots, 13,14\}$ defined as $\mathrm{R}=\{(x, y): 3 x-y=0\}$

Given a non empty set $X$, consider $P ( X )$ which is the set of all subsets of $X$.

Define the relation $R$ in $P(X)$ as follows :

For subsets $A,\, B$ in $P(X),$ $ARB$ if and only if $A \subset B .$ Is $R$ an equivalence relation on $P ( X ) ?$ Justify your answer.