Let the universal set \mathcal{U} = \{x \in \mathbb{Z}^+ : x \le 60\}. Subsets A, B, and C… — JAMB UTME

Let the universal set $\mathcal{U} = \{x \in \mathbb{Z}^+ : x \le 60\}$ . Subsets $A$ , $B$ , and $C$ of $\mathcal{U}$ are defined as:

$A = \{x \in \mathcal{U} : x \text{ is a multiple of } 3\}$
$B = \{x \in \mathcal{U} : x \text{ is a multiple of } 4\}$
$C = \{x \in \mathcal{U} : x \text{ is a factor of } 60\}$

What is the cardinality of the set $(A \Delta B) \setminus C'$ , where $A \Delta B$ represents the symmetric difference of $A$ and $B$ , and $C'$ is the complement of $C$ in $\mathcal{U}$ ?

6Answer
B
8
C
12
D
19

Answer

Using the set identity

X \setminus Y' = X \cap Y

, the expression

(A \Delta B) \setminus C'

simplifies directly to

(A \Delta B) \cap C

. The set

C

consists of all factors of 60:

\{1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\}

. The symmetric difference

A \Delta B

consists of numbers that are divisible by 3 or 4, but not both (excluding multiples of 12). Checking the 12 factors of 60 against this rule, the numbers 3, 6, 15, and 30 are multiples of 3 but not 4, while 4 and 20 are multiples of 4 but not 3. The numbers 12 and 60 are multiples of both 3 and 4, so they are excluded. Thus, the resulting set is

\{3, 4, 6, 15, 20, 30\}

, which contains exactly 6 elements.

Step-by-Step Solution

Simplify the set expression using complement and relative difference identities.

(A \Delta B) \setminus C' = (A \Delta B) \cap (C')' = (A \Delta B) \cap C

By definition of set difference,

X \setminus Y = X \cap Y'

. Replacing

Y

with

C'

gives

X \cap (C')' = X \cap C

List all elements in set

C

(factors of 60).

C = \{1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\}

60 prime factorizes as

2^2 \times 3^1 \times 5^1

, giving

(2+1)(1+1)(1+1) = 12

positive factors.

Recall the definition of symmetric difference

A \Delta B

A \Delta B = (A \cup B) \setminus (A \cap B)

An element belongs to

A \Delta B

if it is a multiple of 3 or a multiple of 4, but NOT a multiple of both 3 and 4 (i.e. not a multiple of 12).

Evaluate which elements of

C

belong to

A \Delta B

The elements in

C

that are multiples of 3 only (not 12) are

\{3, 6, 15, 30\}

. The elements in

C

that are multiples of 4 only (not 12) are

\{4, 20\}

. Elements

\{12, 60\}

are multiples of 12, so they are excluded.

Testing each factor of 60 against the criteria for symmetric difference.

Form the set

(A \Delta B) \cap C

and find its cardinality.

(A \Delta B) \cap C = \{3, 4, 6, 15, 20, 30\}

, so

n((A \Delta B) \cap C) = 6

Counting the elements in the intersection.

Key Concept

Symmetric Difference and Set Complement Identities

Estimated Time:2m 30s

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