Union Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Union.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

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.

Common stuck point: Union doesn't duplicate—element 2 appears once in the result.

Sense of Study hint: Write out both sets' elements side by side, then merge them into one list, crossing off any repeats.

easy

Let A = \{1, 3, 5\} and B = \{2, 3, 4\}. Find A \cup B.

1
Recall the definition: A \cup B = \{x : x \in A \text{ or } x \in B\}. The word 'or' is inclusive — an element belongs to the union if it appears in at least one of the sets.
2
List all elements from A = \{1,3,5\} and B = \{2,3,4\}, including each at most once: 1 (from A), 2 (from B), 3 (in both), 4 (from B), 5 (from A).
3
Therefore A \cup B = \{1,2,3,4,5\}. Notice |A \cup B| = 5 = |A| + |B| - |A \cap B| = 3 + 3 - 1, confirming the inclusion-exclusion principle.

Answer

A \cup B = \{1, 2, 3, 4, 5\}

The union operator collects all elements from both sets. Duplicate elements are listed only once because sets contain distinct elements.

medium

Let A = \{x \in \mathbb{R} : x > 2\} and B = \{x \in \mathbb{R} : x < 5\}. Express A \cup B in interval notation.

Try these problems on your own first, then open the solution to compare your method.

easy

Let P = \{a, b, c\} and Q = \{c, d, e\}. Find P \cup Q.

easy

Find A \cup B if A = \{a, b, d\} and B = \{b, c, d, e\}.

These ideas may be useful before you work through the harder examples.

set