The union of sets A and B is the set of all elements that belong to A, to B, or to both, written A \cup B. Union combines overlapping categories without double-counting, a tool used in probability, data analysis, and every branch of math.
Definition
The union of sets A and B is the set of all elements that belong to A, to B, or to both, written A \cup B.
💡 Intuition
Pour both sets into one container and remove duplicates. Everything from either pile ends up in the union — this is the OR operation for sets.
🎯 Core Idea
x \in A \cup B if and only if x \in A OR x \in B. Union corresponds exactly to logical OR.
Example
Formula
Notation
A \cup B
🌟 Why It Matters
Union combines overlapping categories without double-counting, a tool used in probability, data analysis, and every branch of math.
💭 Hint When Stuck
Write out both sets' elements side by side, then merge them into one list, crossing off any repeats.
Formal View
Related Concepts
🚧 Common Stuck Point
Union doesn't duplicate—element 2 appears once in the result.
⚠️ Common Mistakes
- Confusing union (\cup) with intersection (\cap) — union includes ALL elements from both sets
- Including duplicate elements in the result — sets never have duplicates, so \{1, 2\} \cup \{2, 3\} = \{1, 2, 3\}, not \{1, 2, 2, 3\}
- Forgetting that A \cup \emptyset = A, not \emptyset
Go Deeper
Frequently Asked Questions
What is Union in Math?
The union of sets A and B is the set of all elements that belong to A, to B, or to both, written A \cup B.
What is the Union formula?
A \cup B = \{x : x \in A \text{ or } x \in B\}
When do you use Union?
Write out both sets' elements side by side, then merge them into one list, crossing off any repeats.
Prerequisites
Next Steps
Cross-Subject Connections
How Union Connects to Other Ideas
To understand union, you should first be comfortable with set. Once you have a solid grasp of union, you can move on to intersection and venn diagram.
Visualization
StaticVisual representation of Union