Question

# Show that each of the relation R in the set A={x∈Z:0≤x≤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.

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Solution

## Given that A={x∈Z:0≤x≤12} ={0,1,2,3,4,5,6,7,8,9,10,11,12} R={(a,b):a=b} For any element a∈A, we have (a,a)∈R, as a =a. Therefore, R is reflexive. Now, let (a,b)∈R⇒(b,a)∈R ⇒a=b⇒b=a⇒(b,a)∈R Therefore, R is symmetric. Now, let (a,b)∈R and (b,c)∈R ⇒a=b and b=c⇒(a,c)∈R. Therefore, R is transitive. Hence, R is an equivalence relation. The elements in R that are related to 1 will be those elements from set A. Which are equal to 1 Hence, the set of elements related to 1 is {1}.

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