How many distinct equivalence relations can be defined on the set $S=\{a,b,c\}$?

Step-by-step solution

1. An equivalence relation corresponds to a partition of the underlying set.
2. List all possible partitions of a 3‑element set:
3. 1) \{\{a\},\{b\},\{c\}\} (three singletons).
4. 2) \{\{a,b\},\{c\}\}.
5. 3) \{\{a,c\},\{b\}\}.
6. 4) \{\{b,c\},\{a\}\}.
7. 5) \{\{a,b,c\}\} (the whole set as one block).
8. Each partition determines exactly one equivalence relation, so the number of equivalence relations equals the number of partitions.
9. There are 5 partitions; therefore there are 5 distinct equivalence relations on $S$.

Concept tested: Counting equivalence relations via partitions

Published 2 September 2026 · How this solution was prepared and checked

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