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Question

Show that each of the relation R in the set A={xZ:0x12}, 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={xZ:0x12}
={0,1,2,3,4,5,6,7,8,9,10,11,12}

R={(a,b):a=b}
For any element aA, we have (a,a)R, as a =a.
Therefore, R is reflexive.
Now, let (a,b)R(b,a)R
a=bb=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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