Exercise 1166Logic, sets and proofs › RelationsL1Added 3 Nov 2026

Partitions induce equivalence relations

#partition#equivalence#converse

Statement

Show that any partition of a set $A$ defines an equivalence relation whose classes are the blocks of the partition.

Hint

Define $a\sim b$ iff $a$ and $b$ are in the same block.

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Solution

Definition. $a\sim b\iff\exists$ block $B$ with $a,b\in B$ .

Reflexive. $a\in$ some block $B$ , so $a\sim a$ .

Symmetric. If $a,b\in B$ , then $b,a\in B$ .

Transitive. If $a,b\in B$ and $b,c\in B'$ , then $b\in B\cap B'$ . Since blocks are disjoint, $B=B'$ (as $b$ belongs to both), so $a,c\in B$ and $a\sim c$ .

Conclusion. The partition induces an equivalence relation. This gives a bijection between equivalence relations and partitions.

Partitions induce equivalence relations

Statement

Solution

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