Let the universal set be \mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 20\}. Consider two… — JAMB UTME

Let the universal set be $\mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 20\}$ . Consider two subsets of $\mathcal{U}$ given by $A = \{x \in \mathcal{U} : x \text{ is a multiple of } 4\}$ and $B = \{x \in \mathcal{U} : x \text{ is a factor of } 20\}$ . What is the number of elements in the complement of $(A \cup B)$ , denoted as $n((A \cup B)')$ ?

A
9
11Cevap
C
13
D
15

Cevap

The universal set contains 20 elements. The set of multiples of 4 within this range has 5 elements, and the set of factors of 20 has 6 elements. Their shared elements are 4 and 20 (2 elements). Thus, the union contains

5 + 6 - 2 = 9

elements. Subtracting this from 20 gives 11 elements in the complement.

Adım Adım Çözüm

Identify the elements of the universal set and the subsets

A

and

B

\mathcal{U} = \{1, 2, 3, \dots, 20\}

with

n(\mathcal{U}) = 20

A = \{4, 8, 12, 16, 20\}

with

n(A) = 5

B = \{1, 2, 4, 5, 10, 20\}

with

n(B) = 6

Listing the elements clearly determines the cardinality of each individual set.

Find the intersection and union of sets

A

and

B

A \cap B = \{4, 20\}

, so

n(A \cap B) = 2

A \cup B = \{1, 2, 4, 5, 8, 10, 12, 16, 20\}

, so

n(A \cup B) = 5 + 6 - 2 = 9

Applying the principle of inclusion-exclusion avoids double-counting common elements.

Calculate the cardinality of the complement

(A \cup B)'

n((A \cup B)') = n(\mathcal{U}) - n(A \cup B) = 20 - 9 = 11

The complement contains all elements in the universal set that are not in the union.

Anahtar Kavram

Set Complements and Inclusion-Exclusion Principle

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru