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Question

Show that each of the relation R in the set A={xZ:0x12}, given by is an equivalence relation. Find the set of all elements related to 1 in each case.
R={(a,b):a=b}

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Solution

A={xZ:0x12}={0,1,2,3,4,......,11,12}
R={(a,b):a=b}={(0,0),(1,1),(2,2),........,(11,11),(12,12)}
For any element aA, we have (a,a)R, since a=a.
R is reflexive.
Now, let (a,b)R.
a=b
b=a
(b,a)R
R is symmetric.
Now, let (a,b)R and (b,c)R.
a=b and b=c
a=c
(a,c)R
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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