Let the universal set be \mathcal{U} = \{x \in \mathbb{Z}^+ : 1 \le x \le 15\}. Consider t… — JAMB UTME

Let the universal set be $\mathcal{U} = \{x \in \mathbb{Z}^+ : 1 \le x \le 15\}$ . Consider two subsets of $\mathcal{U}$ defined as $A = \{x \in \mathcal{U} : x \text{ is prime}\}$ and $B = \{x \in \mathcal{U} : x \text{ is an odd integer greater than } 1\}$ . What is the number of elements in $(A \cap B)' \setminus (A \cup B)'$ ?

$3$ Answer
B
$5$
C
$7$
D
$10$

Answer

Evaluating

(A \cap B)'

gives

\{1, 2, 4, 6, 8, 9, 10, 12, 14, 15\}

and evaluating

(A \cup B)'

gives

\{1, 4, 6, 8, 10, 12, 14\}

. Subtracting

(A \cup B)'

from

(A \cap B)'

leaves

\{2, 9, 15\}

, which has 3 elements.

Step-by-Step Solution

List the elements of the universal set

\mathcal{U}

and subsets

A

and

B

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

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

, and

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

Explicitly writing the elements helps ensure set operations are performed accurately.

Find the intersection

A \cap B

and union

A \cup B

A \cap B = \{3, 5, 7, 11, 13\}

and

A \cup B = \{2, 3, 5, 7, 9, 11, 13, 15\}

These intermediate set operations are required to evaluate their complements.

Determine the complements

(A \cap B)'

and

(A \cup B)'

relative to

\mathcal{U}

(A \cap B)' = \{1, 2, 4, 6, 8, 9, 10, 12, 14, 15\}

and

(A \cup B)' = \{1, 4, 6, 8, 10, 12, 14\}

The complement of a set consists of all elements in

\mathcal{U}

not present in that set.

Calculate the set difference

(A \cap B)' \setminus (A \cup B)'

and count its elements.

(A \cap B)' \setminus (A \cup B)' = \{2, 9, 15\}

, which contains 3 elements.

Set difference removes all elements of

(A \cup B)'

from

(A \cap B)'

, leaving elements present in

A \cup B

but not in

A \cap B

Key Concept

Set Complements and Relative Difference

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic