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Given the equivalence relation below, describe the partition?

Equivalence relation: $$R=\{(a,a),(b,b), (c,c), (d,d), (e,e), (a,b), (b,a), (a,d), (d,a), (b,d), (d,b)\}$$ on $S=\{a,b,c,d,e\}$.

I'm not sure what exactly the question is asking or how to format the answer. So far, I think the partition depends on if the pair has the same two elements or if the pair contains an $a$,$b$ or $d$. Thanks

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    What is the definition you have that relates equivalence relations on $S$ and equivalence classes/partitions of $S$? – 2017-01-23

2 Answers 2

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We know that any equivalence relation gives a rise to a partition of the set. The partition is given in such a manner that $a$ and $b$ are in the same subset iff $a\sim b$. Hence from the relation we can conclude that the partition is:

$$S_1 = \{a,b,d\} \quad S_2 = \{c\} \quad S_3 = \{e\}$$

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A relation is (by definition) fully defined by its graph. So, here the subset R of SxS. As you may know S/R, the collection of all of the equivalence classes, is a partition of S. So, what are the equivalence classes? Well, we have: [a] = {a, b, d}. Hence, [a]=[b]=[d]. For c, we have: [c] = {c}, and for e: [e] = {e}. So, the partition of S associated with R is S/R = {{a, b, d}, {c}, {e}}.