Given the universal set U = \{x \in \mathbb{Z} : 1 \le x \le 20\}. Let P = \{x \in U : x \… — JAMB UTME

Given the universal set $U = \{x \in \mathbb{Z} : 1 \le x \le 20\}$ . Let $P = \{x \in U : x \text{ is a prime number}\}$ , $Q = \{x \in U : x \text{ is an odd integer}\}$ , and $R = \{x \in U : x \text{ is a multiple of } 3\}$ . What is the value of $n((P \cup Q) \cap R')$ ?

A
7
8Answer
C
11
D
14

Answer

The correct value is 8. The union

P \cup Q

yields

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

. Intersecting this set with

R'

means removing any element that is a multiple of 3. The multiples of 3 in

P \cup Q

are 3, 9, and 15. Removing these 3 elements from the 11 elements of

P \cup Q

leaves exactly 8 elements.

Step-by-Step Solution

Identify the elements of sets P, Q, and R within the universal set U.

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

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

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

, and

R = \{3, 6, 9, 12, 15, 18\}

Listing elements helps accurately compute set unions and intersections.

Find the union of sets P and Q, denoted as P ∪ Q.

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

, so

n(P \cup Q) = 11

Combining elements of both P and Q without repetition gives their union.

Find the complement of set R with respect to U, denoted as R'.

R' = \{1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20\}

The complement set R' contains all elements in U that are not multiples of 3.

Determine the intersection of (P ∪ Q) and R'.

(P \cup Q) \cap R' = \{1, 2, 5, 7, 11, 13, 17, 19\}

. The number of elements is 8.

This removes the multiples of 3 (namely 3, 9, and 15) from the set

P \cup Q

Key Concept

Set operations including union, intersection, and set complementation.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic