How can I prove that $R$ is an equivalence relation in $A$?

Question

Given the set $A = \{10, 11, 12,. . . , 30\}$ containing the integers between $10$ and $30$,

consider the relationship $R$:

$$x R y \iff \text{the first digit of } x \text{ is equal to the first digit of } y.$$

How can I prove that $R$ is an equivalence relation in $A$?

Answer 1

First of all , an equivalence relation is a relation that is transitive, symmetric as well as reflexive. Also a relation on A means that the relation R is from the set A to the set A itself.

First, the first digit of any number is obviously equal to its own first digit, hence the relation is reflexive.

Secondly, Consider 2 elements whose first digits are same , say a and b. Again, it is clear that the first digit of b and a are also same, hence it is a symmetric relation

Lastly, Suppose 2 elements a and b have the same first digit and so do two elements b and c. Clearly, the first digit of a and c are also equal each being equal to the first digit of b. Hence, R is transitive

Finally since the relation is symmetric, transitive as well as reflexive, the relation R is an equivalence relation

Answer 2

This relation is associated to the partition of the given set: $$ \{10,11,12,\dots,30\}= \{10,11,\dots,19\}\cup\{20,21,\dots,29\}\cup\{30\}. $$