-1
$\begingroup$

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 $A$?

$\endgroup$

closed as off-topic by José Carlos Santos, ahulpke, Sahiba Arora, Arnaud Mortier, Dietrich Burde Feb 11 '18 at 20:17

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, ahulpke, Sahiba Arora, Arnaud Mortier, Dietrich Burde
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ What did you try? $\endgroup$ – José Carlos Santos Feb 11 '18 at 10:51
  • 2
    $\begingroup$ By showing R is reflexive, symmetric and transitive. How could it be any simpler? $\endgroup$ – William Elliot Feb 11 '18 at 11:00
1
$\begingroup$

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

$\endgroup$
  • $\begingroup$ The partition of $A$ given by the relation $R$ is $(10,11,12,13,14,15,16,17,18,19),(20,21,22,23,24,25,26,27,28,29),(30)$? $\endgroup$ – user530133 Feb 11 '18 at 11:10
0
$\begingroup$

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

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.