# NCERT Solutions of Class 12 Maths | CBSE Textbook Solutions | Chapter 1 | Relations and Functions | Exercise 1.1 | Question 8 |

Question Details
 Board CBSE Book NCERT Textbook Class 12 Subject Mathematics Chapter 1 [Relations and Functions] Exercise 1.1 Question No. 8 Question Type Exercise

Question: Show that the relation $$\mathrm{R}$$ in the set $$\mathrm{A}=\{1,2,3,4,5\}$$ given by $$\mathrm{R}=\{(a, b):|a-b|$$ is even $$\}$$, is an equivalence relation. Show that all the elements of $$\{1,3,5\}$$ are related to each other and all the elements of $$\{2,4\}$$ are related to each other. But no element of $$\{1,3,5\}$$ is related to any element of $$\{2,4\}$$.

Given $$A=\{1,2,3,4,5\}$$ and $$R=\{(a, b):|a-b|$$ is even $$\}$$

(a) Reflexive : $$R=\{(a, a):|a-a|=0$$ zero is an even no. $$\}$$
(b) Symmetric : If $$R=\left\{\left(a_1, a_2\right):\left|a_1-a_2\right|\right.$$ is even is true $$\}$$ then $$R=\left\{\left(a_2, a_1\right):\left|a_2-a_1\right|\right.$$ will be even $$\}$$
So, symmetric.

(c) Transitive : If $$R=\left\{\left(a_1, a_2\right)\right.$$ : $$\left|a_1-a_2\right|$$ is even $$\}$$ and $$R=\left\{\left(a_2, a_3\right):\left|a_2-a_3\right|\right.$$ is even $$\}$$ then $$R=\left\{\left(a_1, a_3\right)\right\}$$ will also be even. So equivalence relation.
This is nothing but concept of equivalence class. $$\{1,3,5\}$$ are odd numbers such that difference is always even. Thus $$\{(1,3)\}$$, $$\{(1,5)\}$$ will be even. So this is an equivalence class of 1 denoted by [1]. Now $$\{2,4\}$$ are even numbers here also difference is even. These two are disjoint as well as $$\{1,3,5\} \cup\{2,4\} \equiv\{1,2,3,4,5\} \equiv A$$.

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