Let the universal set be \mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 12\}. Consider the… — JAMB UTME

Let the universal set be $\mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 12\}$ . Consider the subsets $P = \{x : x \text{ is a factor of } 12\}$ and $Q = \{x : x \text{ is an even number}, 1 \le x \le 12\}$ . Which of the following sets represents $(P \cup Q)'$ ?

$\{5, 7, 9, 11\}$ Answer
B
$\{2, 4, 6, 12\}$
C
$\{5, 7, 11\}$
D
$\{1, 3, 5, 7, 9, 11\}$

Answer

\{5, 7, 9, 11\}

The set of factors of 12 within the given range is {1, 2, 3, 4, 6, 12} and the set of even numbers up to 12 is {2, 4, 6, 8, 10, 12}. Their union contains all numbers that are either even or factors of 12, which is {1, 2, 3, 4, 6, 8, 10, 12}. Subtracting this union from the universal set {1, 2, ..., 12} yields {5, 7, 9, 11}.

Step-by-Step Solution

List all elements of the universal set and the subsets

P

and

Q

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

P = \{1, 2, 3, 4, 6, 12\}

, and

Q = \{2, 4, 6, 8, 10, 12\}

Identify the element listings explicitly from the given definitions.

Find the union of sets

P

and

Q

, denoted by

P \cup Q

P \cup Q = \{1, 2, 3, 4, 6, 8, 10, 12\}

Combine all unique elements belonging to either

P

Q

, or both.

Find the complement

(P \cup Q)'

relative to the universal set

\mathcal{U}

(P \cup Q)' = \mathcal{U} \setminus (P \cup Q) = \{5, 7, 9, 11\}

Select all elements in

\mathcal{U}

that do not appear in

P \cup Q

Key Concept

Complement of a Set Union

Estimated Time:50s

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