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Difficulty: MediumSets and Set Operations

Given the universal set U={x∈Z:1≤x≤30}\mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 30\}, let A={x∈U:x is a multiple of 4}A = \{x \in \mathcal{U} : x \text{ is a multiple of } 4\} and B={x∈U:x is a perfect square}B = \{x \in \mathcal{U} : x \text{ is a perfect square}\}. What is the number of elements in (A∪B)′(A \cup B)'?

  1. 20Answer
  2. B
    18
  3. C
    10
  4. D
    22

Answer

The cardinality of (A∪B)′(A \cup B)' is 20.
The universal set has 30 elements. Set AA has 7 elements (multiples of 4 up to 30) and set BB has 5 elements (perfect squares up to 30). The numbers 4 and 16 belong to both sets, so the intersection has 2 elements. By the principle of inclusion-exclusion, the union A∪BA \cup B contains 7+5−2=107 + 5 - 2 = 10 elements. The complement (A∪B)′(A \cup B)' contains all elements in the universal set that are not in the union, giving 30−10=2030 - 10 = 20 elements.

Step-by-Step Solution

1
Determine the elements and cardinality of the universal set U\mathcal{U}.
U={1,2,3,…,30}\mathcal{U} = \{1, 2, 3, \dots, 30\}, so n(U)=30n(\mathcal{U}) = 30.
The universal set bounds all possible elements under consideration.
2
List the elements of set AA and set BB, and find their individual cardinalities.
A={4,8,12,16,20,24,28}  ⟹  n(A)=7A = \{4, 8, 12, 16, 20, 24, 28\} \implies n(A) = 7.
B={1,4,9,16,25}  ⟹  n(B)=5B = \{1, 4, 9, 16, 25\} \implies n(B) = 5.
Identify multiples of 4 and perfect squares within the range 1 to 30.
3
Find the intersection A∩BA \cap B and compute the cardinality of the union A∪BA \cup B.
A∩B={4,16}  ⟹  n(A∩B)=2A \cap B = \{4, 16\} \implies n(A \cap B) = 2.
n(A∪B)=n(A)+n(B)−n(A∩B)=7+5−2=10n(A \cup B) = n(A) + n(B) - n(A \cap B) = 7 + 5 - 2 = 10.
Apply the principle of inclusion-exclusion to avoid double-counting elements belonging to both sets.
4
Calculate the cardinality of the complement (A∪B)′(A \cup B)'.
n((A∪B)′)=n(U)−n(A∪B)=30−10=20n((A \cup B)') = n(\mathcal{U}) - n(A \cup B) = 30 - 10 = 20.
The complement set consists of all elements in the universal set that are not in A∪BA \cup B.

Key Concept

Complement of Set Union and Inclusion-Exclusion Principle
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Topics

Sets and Set Operations

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