Union and Intersection

Set Theory · Axiom Academy

Combining sets in two fundamental ways Just as we can add and multiply numbers, we can combine sets using operations. The two most fundamental set operations are union and intersection . Union: Everything that's in either set (or both) Intersection: Only what's in both sets Venn diagram showing two overlapping sets The union of sets A and B, written , is the set of all elements that belong to A or B (or both). Note: Each element appears only once in the union, even if it's in both sets! An element if and only if or (or both). The intersection of sets A and B, written , is the set of all elements that belong to both A and B. Only 3 and 4 are in both sets. When two sets have no elements in common, their intersection is the empty set. We say the sets are disjoint . Let's compare these operations side by side: Union and intersection satisfy several important algebraic laws: Unlike regular arithmetic, both distributive laws hold for sets! In numbers, , but for sets, both distributions work. Union and intersection interact with subsets in useful ways: The intersection is always "smaller" and the union is always "larger" (in terms of subset containment). For finite sets, the sizes are related by: We subtract the intersection to avoid counting shared elements twice! Union (∪) : Combines all elements from both sets Intersection (∩) : Keeps only shared elements Both operations are commutative and associative Both distributive laws hold for sets Use inclusion-exclusion for counting

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