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.
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 Questions
- A = {2, 4, 6, 8} and B = {1, 2, 3, 4}. Find A ∪ B and |A ∪ B|.
- C = {multiples of 3 up to 15} and D = {multiples of 5 up to 15}. Find C ∪ D.
- In a group of 50 people, 30 like tea and 25 like coffee. 10 like both. How many like at least one?
- E = {vowels} and F = {a, b, c, d, e}. Find E ∪ F.
- If |A| = 12, |B| = 9, and |A ∪ B| = 17, find |A ∩ B|.