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Show that each of the relation R in the set A = {x belongs to Z : 0 is less than or equal to x is less than or equal to 12}, given by R = {(a, b) : a = b} is an equivalence relation. Find the set of all elements related to 1 in each case.

Q.9 Show that each of the relation R in the set A = \{x \in Z : 0 \leq x \leq 12\}, given by

(ii)  R = \{(a, b) : a = b\} is an equivalence relation. Find the set of all elements related to 1 in each case.

Answers (1)
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A = \{x \in Z : 0 \leq x \leq 12\}

A=\left \{ 0,1,2,3,4,5,6,7,8,9,10,11,12 \right \}

R = \{(a, b) : a = b\}

For a\in A , (a,a)\in R  as  a=a

Henec, it is reflexive.

Let, (a,b)\in R i.e. a=b 

 a=b \Rightarrow  b=a  i.e.(b,a)\in R

Hence, it is symmetric.

Let, (a,b)\in R i.e. a=b   and    (b,c)\in R  i.e. b=c

       \therefore  a=b=c

a=c i.e. (a,c)\in R

Hence, it is transitive.

Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.

The set of all elements related to 1 is {1}

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