# 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.

We have,

A = {x Z : 0 ≤ x ≤ 12} be a set and

R = {(a, b) : a = b} be a relation on A

Now,

Proof :

To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.

Reflexivity : For Reflexivity, we need to prove that-

(a, a) R

Let a A

a = a

(a, a) R

R is reflexive

Symmetric : For Symmetric, we need to prove that-

If (a, b) R, then (b, a) R

Let a, b A and (a, b) R

a = b

b = a

(b, a) R

R is symmetric

Transitive : : For Transitivity, we need to prove that-

If (a, b) R and (b, c) R, then (a, c) R

Let a, b & c A

and Let (a, b) R and (b, c) R

a = b and b = c

a = c

(a, c) R

R is transitive

Since, R is being reflexive, symmetric and transitive, so R is an equivalence relation.

Also, we need to find the set of all elements related to 1.

Since the relation is given by, R = {(a, b) : a = b}, and 1 is an element of A,

R = {(1, 1) : 1 = 1}

Thus, the set of all element related to 1 is 1.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Functions - 0152 mins
Range of Functions58 mins
Quick Revision of Types of Relations59 mins
Some standard real functions61 mins
Battle of Graphs | various functions & their Graphs48 mins
Functions - 0947 mins
Quick Recap lecture of important graphs & functions58 mins