Intersections & unions of equivalence relations
The intersection of any family of equivalence relations is again an equivalence relation — each of the three properties is preserved by intersection, which is exactly why closures exist. The union is different: it stays reflexive and symmetric but usually loses transitivity, so the smallest equivalence relation containing both is the equivalence closure of the union, not the union itself.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Prove intersection closure property by propertyReflexive: (a,a) is in each, so in the intersection. Symmetric and transitive: the hypotheses hold in each relation, so the conclusions do too.
- Describe the intersection’s classesThe class of a under ∼₁ ∩ ∼₂ is [a]₁ ∩ [a]₂ — the blocks are the pairwise intersections of blocks.
- Test the union for transitivityLook for a ∼₁ b and b ∼₂ c with a and c unrelated in both. This is the standard counterexample shape.
- Take the closure when you need a joinThe equivalence closure of ∼₁ ∪ ∼₂ merges any blocks that overlap, repeatedly, until stable.
Worked example
On {1,2,3} let ∼₁ have classes {1,2},{3} and ∼₂ have classes {1},{2,3}. Compute the intersection and show the union is not transitive.
- Intersection blocks are pairwise intersections: {1,2}∩{1} = {1}, {1,2}∩{2,3} = {2}, {3}∩{2,3} = {3}.
- So the intersection is equality, with classes {1},{2},{3}.
- Union contains (1,2) from ∼₁ and (2,3) from ∼₂.
- Transitivity would need (1,3), which is in neither relation.
Answer. The intersection is equality; the union contains (1,2) and (2,3) but not (1,3), so it is not transitive.
Where marks get dropped
These are the specific errors that cost credit on intersections & unions of equivalence relations questions — QED's rubric penalises each of them separately.
- Assuming the union of two equivalence relations is an equivalence relation. It is only when one contains the other.
- Computing intersection blocks by intersecting the wrong pairs. Each block of the result is [a]₁ ∩ [a]₂ for a common element a — empty intersections do not appear.
- Forgetting the result of intersecting the classes can be equality, the finest partition, as it is above.
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Intersections & unions of equivalence relations — frequently asked questions
When IS the union an equivalence relation?
Exactly when one relation contains the other, so the union is the larger one. Otherwise some pair of blocks overlaps in a way that breaks transitivity.
Why does intersection always work?
Because reflexivity, symmetry and transitivity are all universally quantified Horn conditions, and such conditions are preserved by arbitrary intersections. That is the general reason closures exist for them.
What partition does the intersection give?
The common refinement: blocks are the non-empty intersections of blocks from the two partitions. It is the meet in the lattice of equivalence relations.
The rest of Equivalence Relations
Equivalence classes, partitions and quotient sets. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Verifying an equivalence relation
- 2Equivalence classes [a]
- 3The class–partition correspondence
- 4The quotient set A/∼
- 5Congruence mod n
- 6The kernel of a function as an equivalence
- 7Well-definedness of operations on classes
- 8Counting equivalence relations
- 9Equivalence closure of a relation
- 10Refinement of equivalence relations
- 11Intersections & unions of equivalence relations
- 12Bell numbers & counting partitions
- 13Isomorphism & similarity as equivalences
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