Union of Sets - Everything in Either Set
The union of two sets is a new set containing every element that belongs to either set (or both). Think of it as combining two groups together – every member of either group is included in the result, but no one is counted twice.
The ∪ symbol was introduced by the Italian mathematician Giuseppe Peano in 1888, in his book Calcolo Geometrico, as part of his broader project to build mathematics on a small set of precise symbols and axioms – the same project that gave us the Peano axioms for the natural numbers. The counting rule behind union, the inclusion-exclusion principle, is much older: it traces back to the French mathematician Abraham de Moivre, who used an early version of it in his 1718 book on probability, The Doctrine of Chances, to avoid double-counting overlapping outcomes when calculating the odds of games of chance.
Union shows up constantly outside the classroom too: a spreadsheet filter for “sales in the UK or sales over £500” is a union of two conditions, and search engines combine result sets with union logic whenever you search for “cats OR dogs”.
Notation and Definition
A ∪ B = {x : x ∈ A or x ∈ B}
The symbol ∪ is called the cup symbol. It stands for union. An element belongs to A ∪ B if it is in A, in B, or in both.
Key Properties of Union
| Property | Meaning | Example |
|---|---|---|
| Commutative | A ∪ B = B ∪ A | Order of the sets does not matter |
| Associative | (A ∪ B) ∪ C = A ∪ (B ∪ C) | Grouping does not matter |
| Identity | A ∪ ∅ = A | Unioning with the empty set changes nothing |
| Idempotent | A ∪ A = A | Unioning a set with itself gives the same set |
| Universal | A ∪ ξ = ξ | Unioning with the universal set gives the universal set |
Cardinality of a Union
The inclusion-exclusion principle gives the size of A ∪ B:
|A ∪ B| = |A| + |B| − |A ∩ B|
Subtracting |A ∩ B| prevents elements in both sets from being counted twice.
Worked Examples
A ∪ B = {1, 2, 3, 4, 5, 6}. |A ∪ B| = 6.
Using the formula: |A| + |B| − |A ∩ B| = 4 + 4 − 2 = 6. ✓
P = {R, A, T, E}, Q = {T, E, A, R}. Both sets contain the same letters in different order.
P ∪ Q = {R, A, T, E}. (P = Q here, so the union equals either set.)
|F ∪ B| = 18 + 15 − 8 = 25 students.
A = {2, 3, 5, 7}. B = {1, 3, 5, 7, 9}.
A ∪ B = {1, 2, 3, 5, 7, 9}. |A ∪ B| = 6.
Key Takeaways
- A ∪ B contains every element in A, B, or both – no duplicates.
- Union is commutative (A ∪ B = B ∪ A) and associative.
- |A ∪ B| = |A| + |B| − |A ∩ B| (inclusion-exclusion principle).
- A ∪ ∅ = A; A ∪ ξ = ξ.
Practice: Union Cardinality
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