Given the universal set \mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 30\}, let A = \{x \… — JAMB UTME

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

20Cevap
B
18
C
10
D
22

Cevap

The cardinality of

(A \cup B)'

is 20.

The universal set has 30 elements. Set

A

has 7 elements (multiples of 4 up to 30) and set

B

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 \cup B

contains

7 + 5 - 2 = 10

elements. The complement

(A \cup B)'

contains all elements in the universal set that are not in the union, giving

30 - 10 = 20

elements.

Adım Adım Çözüm

Determine the elements and cardinality of the universal set

\mathcal{U}

\mathcal{U} = \{1, 2, 3, \dots, 30\}

, so

n(\mathcal{U}) = 30

The universal set bounds all possible elements under consideration.

List the elements of set

A

and set

B

, and find their individual cardinalities.

A = \{4, 8, 12, 16, 20, 24, 28\} \implies n(A) = 7

B = \{1, 4, 9, 16, 25\} \implies n(B) = 5

Identify multiples of 4 and perfect squares within the range 1 to 30.

Find the intersection

A \cap B

and compute the cardinality of the union

A \cup B

A \cap B = \{4, 16\} \implies n(A \cap B) = 2

n(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.

Calculate the cardinality of the complement

(A \cup B)'

n((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 \cup B

Anahtar Kavram

Complement of Set Union and Inclusion-Exclusion Principle

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