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Question
Show that the relation R on the set A = {x ∈ Z ; 0 ≤ x ≤ 12}, given by R = {(a, b) : a = b}, is an equivalence relation. Find the set of all elements related to 1.
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Solution
We observe the following properties of R.
Reflexivity : Let a be an arbitrary element of A. Then,
a ∈ R
⇒ a = a [Since, every element is equal to itself]
⇒ (a, a) ∈ R for all a ∈ A
So, R is reflexive on A.
Symmetry : Let (a, b) ∈ R
⇒ a b
⇒ b = a
⇒ (b, a) ∈ R for all a, b ∈ A
So, R is symmetric on A.
Transitivity : Let (a, b) and (b, c) ∈ R
⇒ a =b and b = c
⇒ a = b c
⇒ a = c
⇒ (a, c) ∈ R
So, R is transitive on A.
Hence, R is an equivalence relation on A.
The set of all elements related to 1 is {1}.
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