Union — Definition, Symbol & Examples
Union of Sets
Combining the elements of two or more sets. Union is indicated by the ∪ (cup) symbol.
See also
Key Formula
A∪B={x∣x∈A or x∈B}
Where:
- A,B = Two sets whose elements are being combined
- x = Any element that belongs to at least one of the sets
- ∪ = The union symbol, read as 'union' or 'cup'
Worked Example
Problem: Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. Find A ∪ B.
Step 1: List all elements that are in set A.
A={1,2,3,4}
Step 2: List all elements that are in set B.
B={3,4,5,6}
Step 3: Combine every element from both sets. The elements 3 and 4 appear in both A and B, but you write each element only once in the union.
A∪B={1,2,3,4,5,6}
Step 4: Verify: every element of A is in the result, and every element of B is in the result. No element is listed twice.
Answer: A ∪ B = {1, 2, 3, 4, 5, 6}
Another Example
This example extends union to three sets, showing that the operation chains naturally and that the order does not matter (union is associative).
Problem: Let P = {a, b, c}, Q = {b, c, d}, and R = {c, d, e}. Find P ∪ Q ∪ R.
Step 1: Start by finding P ∪ Q. Combine all elements from P and Q, listing each only once.
P∪Q={a,b,c,d}
Step 2: Now take the result and form its union with R. Add any elements from R that are not already listed.
(P∪Q)∪R={a,b,c,d}∪{c,d,e}
Step 3: The elements c and d are already present, so only e is new.
P∪Q∪R={a,b,c,d,e}
Answer: P ∪ Q ∪ R = {a, b, c, d, e}
Frequently Asked Questions
What is the difference between union and intersection of sets?
Union (A ∪ B) collects every element that is in A or B or both — think of it as combining. Intersection (A ∩ B) keeps only the elements that are in both A and B at the same time — think of it as overlapping. For example, if A = {1, 2, 3} and B = {2, 3, 4}, then A ∪ B = {1, 2, 3, 4} while A ∩ B = {2, 3}.
Can you take the union of a set with the empty set?
Yes. The empty set ∅ contains no elements, so A ∪ ∅ = A for any set A. The empty set is the identity element for union, meaning it does not add or remove anything.
Does the union of two sets ever have fewer elements than either set?
No. The union always contains at least as many elements as the larger of the two sets, because every element of both sets is included. It can, however, have the same number of elements as one set — for instance, if one set is a subset of the other.
Union (A ∪ B) vs. Intersection (A ∩ B)
| Union (A ∪ B) | Intersection (A ∩ B) | |
|---|---|---|
| Definition | All elements in A or B or both | Only elements in both A and B |
| Symbol | ∪ (cup — opens upward) | ∩ (cap — opens downward) |
| Logical equivalent | OR — at least one set must contain x | AND — both sets must contain x |
| Size relationship | |A ∪ B| ≥ max(|A|, |B|) | |A ∩ B| ≤ min(|A|, |B|) |
| Identity element | A ∪ ∅ = A | A ∩ U = A (U is the universal set) |
| Venn diagram area | Entire shaded region of both circles | Only the overlapping region |
Why It Matters
Union appears throughout probability, where P(A ∪ B) represents the probability of event A or event B occurring. It is also fundamental in database queries and computer science, where combining result sets relies on the union operation. You will encounter union regularly in algebra courses when working with solution sets — for instance, the solution to |x| > 3 is expressed as (−∞, −3) ∪ (3, ∞).
Common Mistakes
Mistake: Listing duplicate elements in the union.
Correction: A set never contains duplicate elements. When you form A ∪ B, write each shared element only once. For example, {1, 2} ∪ {2, 3} = {1, 2, 3}, not {1, 2, 2, 3}.
Mistake: Confusing the ∪ (cup) symbol with the ∩ (cap) symbol.
Correction: Remember that ∪ looks like a cup that collects everything (union = 'or'), while ∩ looks like a cap that restricts to what overlaps (intersection = 'and'). A helpful mnemonic: the U in ∪nion opens Upward.
Related Terms
- Set — The fundamental object union operates on
- Intersection — Counterpart operation — elements common to both sets
- Element of a Set — Individual objects that make up a set
- Set Subtraction — Removes elements of one set from another
- Venn Diagrams — Visual tool for illustrating union and intersection