Practice Union in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The union of sets AA and BB is the set of all elements that belong to AA, to BB, or to both, written A∪BA \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.

Showing a random 20 of 50 problems.

Example 1

easy
Let U={1,2,3,4,5}U = \{1, 2, 3, 4, 5\} and A={2,4}A = \{2, 4\}. Find A∪UA \cup U.

Example 2

easy
Compute {1,2}∪{3,4}\{1, 2\} \cup \{3, 4\}.

Example 3

hard
Let A={x∈R:x2≤9}A = \{x \in \mathbb{R} : x^2 \le 9\} and B={x∈R:x≥2}B = \{x \in \mathbb{R} : x \ge 2\}. Express A∪BA \cup B.

Example 4

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

Example 5

easy
Compute ∅∪{4,5,6}\emptyset \cup \{4, 5, 6\}.

Example 6

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

Example 7

medium
Let A={x∈Z:−2≤x≤2}A = \{x \in \mathbb{Z} : -2 \le x \le 2\} and B={x∈Z:0≤x≤4}B = \{x \in \mathbb{Z} : 0 \le x \le 4\}. Find A∪BA \cup B.

Example 8

medium
Compute the union of {1},{2},{3},{4}\{1\}, \{2\}, \{3\}, \{4\} (all four sets together).

Example 9

easy
Is A⊆A∪BA \subseteq A \cup B always true?

Example 10

hard
In a class of 3030 students, 1818 play soccer, 1212 play basketball, and 55 play both. How many play at least one?

Example 11

easy
Is B⊆A∪BB \subseteq A \cup B always true?

Example 12

medium
Let A={A = \{multiples of 33 from 11 to 20}20\} and B={B = \{multiples of 55 from 11 to 20}20\}. List A∪BA \cup B.

Example 13

easy
Compute {1,2}∪{2,1}\{1, 2\} \cup \{2, 1\}.

Example 14

medium
If ∣A∣=5|A| = 5, ∣B∣=3|B| = 3, and ∣A∩B∣=2|A \cap B| = 2, find ∣A∪B∣|A \cup B|.

Example 15

medium
True or false: for any sets A,BA, B, A∪B=B∪AA \cup B = B \cup A.

Example 16

hard
6060 people are surveyed: 4040 like coffee, 3535 like tea, and 55 like neither. How many like at least one of coffee or tea?

Example 17

easy
Compute A∪∅A \cup \emptyset for A={5,6}A = \{5, 6\}.

Example 18

easy
Compute A∪BA \cup B for A={2,4,6,8}A = \{2, 4, 6, 8\} and B={1,3,5}B = \{1, 3, 5\}.

Example 19

challenge
Prove the inclusion-exclusion formula ∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A \cup B| = |A| + |B| - |A \cap B| for finite sets.

Example 20

medium
List A∪B∪CA \cup B \cup C for A={1,2}A=\{1,2\}, B={2,3}B=\{2,3\}, C={3,4}C=\{3,4\}.
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