Let the universal set be \mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 36\}. Subsets A, B… — JAMB UTME

Let the universal set be $\mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 36\}$ . Subsets $A$ , $B$ , and $C$ of $\mathcal{U}$ are defined as follows:
$A = \{x \in \mathcal{U} : x \text{ is a perfect square}\}$ ,
$B = \{x \in \mathcal{U} : x \text{ is a multiple of } 3\}$ , and
$C = \{x \in \mathcal{U} : x \text{ is an even number}\}$ .
What is the cardinality of the set $(A \cup B)' \cap C$ ?

A
8
B
9
10Answer
D
12

Answer

The cardinality of the set

(A \cup B)' \cap C

is 10.

The set

(A \cup B)' \cap C

represents elements in

C

that are neither in

A

nor in

B

. Set

C

contains 18 even integers up to 36. Among these 18 numbers, three are perfect squares (4, 16, 36) and six are multiples of 3 (6, 12, 18, 24, 30, 36). Since 36 satisfies both conditions, the total number of even numbers in

A \cup B

3 + 6 - 1 = 8

. Subtracting these 8 elements from 18 leaves 10 elements.

Step-by-Step Solution

Identify the elements of set C (even numbers in the universal set)

C = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36\}

, so

n(C) = 18

The complement set operation is restricted to elements within set C.

Find the elements of set C that belong to set A (even perfect squares)

C \cap A = \{4, 16, 36\}

, so

n(C \cap A) = 3

Identify which elements in C must be excluded due to being perfect squares.

Find the elements of set C that belong to set B (even multiples of 3, i.e., multiples of 6)

C \cap B = \{6, 12, 18, 24, 30, 36\}

, so

n(C \cap B) = 6

Identify which elements in C must be excluded due to being multiples of 3.

Find the overlap between the excluded sets within C

C \cap A \cap B = \{36\}

, so

n(C \cap A \cap B) = 1

Avoid double-counting 36 when subtracting excluded elements.

Calculate

n(C \cap (A \cup B))

using inclusion-exclusion and find the remaining complement elements in C

n(C \cap (A \cup B)) = 3 + 6 - 1 = 8

. Therefore,

n((A \cup B)' \cap C) = 18 - 8 = 10

Subtracting the 8 excluded even numbers from the total 18 even numbers yields the final cardinality.

Key Concept

Set Complement and Relative Difference with Inclusion-Exclusion

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic