Given the universal set U = \{x \in \mathbb{Z} : 1 \le x \le 20\}, with subsets P = \{x \i… — JAMB UTME

Given the universal set $U = \{x \in \mathbb{Z} : 1 \le x \le 20\}$ , with subsets $P = \{x \in U : x \text{ is a prime number}\}$ and $Q = \{x \in U : x \text{ is an odd integer}\}$ , what is the cardinal number of $(P \cup Q)'$ ?

$9$ Answer
B
$11$
C
$10$
D
$8$

Answer

9

The universal set contains

20

elements. The union

P \cup Q

consists of all odd numbers and prime numbers between

1

and

20

, giving

11

unique elements:

\{1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19\}

. Subtracting these

11

elements from the total

20

elements in

U

gives

n((P \cup Q)') = 20 - 11 = 9

Step-by-Step Solution

Identify the elements of the universal set

U

and subsets

P

and

Q

U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}

, so

n(U) = 20

P = \{2, 3, 5, 7, 11, 13, 17, 19\}

Q = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}

Explicitly listing the set elements helps correctly calculate the union.

Determine the union

P \cup Q

P \cup Q = \{1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19\}

, which contains

11

elements.

The union combines all unique elements present in set

P

, set

Q

, or both.

Calculate the complement set

(P \cup Q)'

and its cardinality.

(P \cup Q)' = U \setminus (P \cup Q) = \{4, 6, 8, 10, 12, 14, 16, 18, 20\}

, so

n((P \cup Q)') = 20 - 11 = 9

The complement of a set contains all elements in the universal set

U

that are not in the given set.

Key Concept

Set Operations and Complement of Sets

Estimated Time:1m 0s

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