I want to determine wether the given relation is an equivalence relation on [$1$, $2$, $3$, $4$, $5$]. And If it is then list all the equivalence classes.
Relation: $\{(1, 1),(2, 2),(3, 3), (4, 4), (5, 5), (1, 5), (5, 1), (3, 5), (5, 3), (1, 3), (3, 1)\}$
My calculations:
It's reflexive since all numbers are related to themselves.
it's symetric.
I'm not sure if it's a transitive set. Since (4, 4), (2, 2) are alone and not connected to any others.
If it is transitive then how would you go about listing the equivalence classes?
Edit: I've gotten feedback now that says that it is transitive. So then I want to list the equivalence classes:
Some examples I have seen have definitions of the equivalence classes but this one doesn't so I assume all I gotta do is:
$[1] = \{ 1, 5, 3\}$
$[2] = \{ 2\}$
$[3] = \{ 3, 5, 1\}$
$[4] = \{ 4\}$
$[5] = \{ 5, 1, 3\}$
or something simillar?
Yes the relation is transitive.
The only way that the relation would not be transitive, would be if there exists an $a,b,c$ such that $(a\sim b$ and $b\sim c),\,$ but $a \not \sim c$.
Equivalence Classes
Careful:
Each class must be pairwise disjoint. In your work, however, note that $[1] = [3]= [5] = \{1, 3, 5\}$
There should be only three equivalence classes: $[1] = \{1, 3, 5\}$, $\;[2] = \{2\},\;$ and $[4] = \{4\}$.
That's essentially what I think you are trying to say, But the relation defined in your question has only 3 unique (and disjoint) classes.