Question:
The number of non-empty equivalence relations on the set \(\{1,2,3\}\) is :
The number of non-empty equivalence relations on the set \(\{1,2,3\}\) is :
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To count the number of equivalence relations, find all the possible partitions of the set.
Updated On: Mar 31, 2026
- 6
- 7
- 5
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The Correct Option is C
Approach Solution - 1
To determine the number of non-empty equivalence relations on the set \(\{1,2,3\}\), we need to find the number of partitions of this set. An equivalence relation corresponds to a partition of the set, and we enumerate the possible partitions of the set \(\{1,2,3\}\) as follows:
- Partition with one subset (trivial partition): \(\{\{1,2,3\}\}\)
- Partitions with two subsets:
- \(\{\{1\},\{2,3\}\}\)
- \(\{\{2\},\{1,3\}\}\)
- \(\{\{3\},\{1,2\}\}\)
- Partition with three subsets: \(\{\{1\},\{2\},\{3\}\}\)
Counting these partitions, we find there are 5 distinct ways to partition the set \(\{1,2,3\}\). Hence, the number of non-empty equivalence relations on the set \(\{1,2,3\}\) is 5.
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Approach Solution -2
Recall: equivalence relations on a set are in one-to-one correspondence with partitions of that set (each equivalence class is a block of the partition).
- So we need the number of partitions of a $3$-element set. This is the Bell number $B_3$.
- List all partitions of $ \{1,2,3\} $ explicitly:
- $\{\{1,2,3\}\}$ — one block (all elements equivalent).
- $\{\{1\},\{2,3\}\}$.
- $\{\{2\},\{1,3\}\}$.
- $\{\{3\},\{1,2\}\}$.
- $\{\{1\},\{2\},\{3\}\}$ — three singleton blocks (only identity pairs).
- Each partition gives exactly one equivalence relation. Hence there are $5$ partitions, so $5$ equivalence relations.
- (If useful, the corresponding relations written as sets of ordered pairs are:)
- For $\{\{1,2,3\}\}$: $\{(i,j): i,j\in\{1,2,3\}\}$ (all $9$ pairs).
- For $\{\{1\},\{2,3\}\}$: $\{(1,1),(2,2),(3,3),(2,3),(3,2)\}$.
- For $\{\{2\},\{1,3\}\}$: $\{(1,1),(2,2),(3,3),(1,3),(3,1)\}$.
- For $\{\{3\},\{1,2\}\}$: $\{(1,1),(2,2),(3,3),(1,2),(2,1)\}$.
- For $\{\{1\},\{2\},\{3\}\}$: $\{(1,1),(2,2),(3,3)\}$ (identity relation).
Answer
The number of equivalence relations on $\{1,2,3\}$ is $5$. (Option 3)
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