Question

Show that each of the relation R in the set, given by (i) (ii) is an equivalence relation. Find the set of all elements related to 1 in each case.

Answer 1

Show that each of 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 in each case.

solution :- given, that the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by
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,a) ∈ R
⇒ a = b
⇒ b = a
⇒ (b,a) ϵ R
Therefore, R is symmetric.
Now, Let (a,b), (b,c) ∈ R
⇒ a = b and b = c
⇒ a = c
⇒ (a,c) ∈ R
therefore, R is transitive.

Therefore, R is an equivalence relation.
The set of elements related to 1 will be those elements from set A which are equal to 1.
Therefore, the set of elements related to 1 is {1}.