Construct an equivalence relation on a given set

Question

can anyone help me on this problem?

I have the set $\{0,1,3,8,9\}$ and I want to define an example of an equivalence relation.

I know that to be an equivalence relation it needs to be reflexive, symmetric and transitive.

I also now that for a set of $5$ elements there are $2^{n^2}$, so if $n=5$ there are $2^{10}$ of relations.

I am not sure how to start because there are $2^{10}$ relations in this set. Do I have to list all of them or is there any other method to find an equivalence relation?

Thank you

Answer 1

Here are some suggestions:

1) Let one number be related to another, if the two numbers are equal. For instance, $3$ is only related to $3$ (itself), and likewise for all other numbers. You can check that this is an equivalence relation.

2) Let one number be related to another, if the two numbers have the same number of holes in them: $0$ has one hole, $1$ has no holes, $3$ has no holes, $8$ has two holes, and $9$ has one hole. For instance, $0$ is related to $9$, because they have the same number of holes.

3) One number is related to another, if the two numbers contain the same number of letters when spelled out. For instance, $0$ is related to $9$, because ZERO and NINE both have four letters.

4) Two numbers are related, if they are both odd, or if they are both even.

Answer 2

Take any partition of $S=\{0,1,3,8,9\}$ for e.g. $A=\{0,1\},B=\{3,8,9\}.$ Define a relation on $S$ by $a\sim b $ iff both $a,b$ belongs to same set in the partition. Then $\sim$ is equivalence relation.

Answer 3

Hint:

There is a one-to-one relation between equivalence relations on a set $S$ and partitions of a set $S$.

Example: $\{\{0,8\},\{1,3,9\}\}$ is a partition of set $\{0,1,3,8,9\}$.

The corresponding equivalence relation is:

$xRy$ iff $x,y$ are both elements of the same set in the partition.

This leads to $0R0$, $0R8$, $1R3$, $3R1$ et cetera.