Given the universal set \mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 15\}, with subsets … — JAMB UTME

Given the universal set $\mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 15\}$ , with subsets $P = \{x \in \mathcal{U} : x \text{ is a prime number}\}$ and $Q = \{x \in \mathcal{U} : x \text{ is an odd number}\}$ , what is the number of elements in the set $(P \cup Q)'$ ?

Answer: 6 / six / 6 elements

Answer

The number of elements in

(P \cup Q)'

is 6.

The set

P \cup Q

consists of all numbers from 1 to 15 that are either prime or odd:

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

. The complement

(P \cup Q)'

relative to

\mathcal{U}

contains all elements of

\mathcal{U}

that are neither prime nor odd, which are the even composite numbers:

\{4, 6, 8, 10, 12, 14\}

. Counting these elements gives 6.

Step-by-Step Solution

List all elements of the universal set

\mathcal{U}

, subset

P

, and subset

Q

\mathcal{U} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}

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

, and

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

Listing the explicit elements allows precise execution of set union and complement operations.

Find the union

P \cup Q

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

The union combines all unique elements that belong to either set

P

, set

Q

, or both.

Determine the complement set

(P \cup Q)'

relative to

\mathcal{U}

and count its elements.

(P \cup Q)' = \{4, 6, 8, 10, 12, 14\}

, which contains 6 elements.

The complement set consists of all elements in

\mathcal{U}

that are not present in

P \cup Q

Key Concept

Complement of Set Union

Question

Answer

Step-by-Step Solution

Key Concept

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