Let the universal set be U = \{x \in \mathbb{Z} : 1 \le x \le 10\}. If P = \{x : x \text{ … — JAMB UTME

Let the universal set be $U = \{x \in \mathbb{Z} : 1 \le x \le 10\}$ . If $P = \{x : x \text{ is a prime number within } U\}$ and $Q = \{x : x \text{ is an even number within } U\}$ , which of the following represents the set $(P \cup Q)'$ ?

A
$\{2\}$
B
$\{3, 5, 7\}$
$\{1, 9\}$ Answer
D
$\{1, 3, 5, 7, 9\}$

Answer

\{1, 9\}

The universal set

U

contains all integers from

1

10

. Prime numbers in this range form

P = \{2, 3, 5, 7\}

, and even numbers form

Q = \{2, 4, 6, 8, 10\}

. The union

P \cup Q

combines all elements from both sets:

\{2, 3, 4, 5, 6, 7, 8, 10\}

. Taking the complement of this union relative to

U

leaves the elements

\{1, 9\}

, making the option stating

\{1, 9\}

correct.

Step-by-Step Solution

List all elements of the universal set

U

and the subsets

P

and

Q

U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}

P = \{2, 3, 5, 7\}

, and

Q = \{2, 4, 6, 8, 10\}

Explicit listing helps identify union and complement elements accurately.

Find the union of sets

P

and

Q

, denoted as

P \cup Q

P \cup Q = \{2, 3, 4, 5, 6, 7, 8, 10\}

The union includes all elements that belong to set

P

, set

Q

, or both.

Determine the complement

(P \cup Q)'

by subtracting

P \cup Q

from the universal set

U

(P \cup Q)' = U \setminus (P \cup Q) = \{1, 9\}

The complement contains all elements of

U

that are not present in

P \cup Q

Key Concept

Set Complement and Union Operations

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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