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Question
Show that the relation R in 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
R = {(a, b) : a = b}
(i) Reflexive:
R = {(0, 0), (1, 1), ..., (12, 12)}
A = {0, 1, 2, ..., 12}
For any element a ∈ A, we have (a, a) ∈ R, since a = a.
∴ R is reflexive.
(ii) Symmetric:
Now, let (a, b) ∈ R.
⇒ a = b
⇒ b = a
⇒ (b, a) ∈ R
∴ R is symmetric.
(iii) Transitive:
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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