Given the universal set \mathcal{E} = \{x \in \mathbb{Z} : 1 \le x \le 20\}, and subsets: … — JAMB UTME

Given the universal set

\mathcal{E} = \{x \in \mathbb{Z} : 1 \le x \le 20\}

, and subsets:

A = \{x \in \mathcal{E} : x \text{ is a multiple of } 3\}

B = \{x \in \mathcal{E} : x \text{ is a multiple of } 4\}

C = \{x \in \mathcal{E} : x \text{ is a prime number}\}

What is the number of elements in the set $(A \cup B)' \cap C$ ?

The universal set contains integers from 1 to 20. Subset

A

has multiples of 3, subset

B

has multiples of 4, and subset

C

contains the prime numbers

\{2, 3, 5, 7, 11, 13, 17, 19\}

. The set

A \cup B

contains all multiples of 3 or 4 within the range, namely

\{3, 4, 6, 8, 9, 12, 15, 16, 18, 20\}

. The complement

(A \cup B)'

consists of elements in the universal set not in

A \cup B

\{1, 2, 5, 7, 10, 11, 13, 14, 17, 19\}

. Taking the intersection of

(A \cup B)'

with

C

filters this list to only the prime numbers:

\{2, 5, 7, 11, 13, 17, 19\}

, giving a cardinality of 7.

List the elements of the universal set and the given subsets.

\mathcal{E} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}

A = \{3, 6, 9, 12, 15, 18\}

B = \{4, 8, 12, 16, 20\}

C = \{2, 3, 5, 7, 11, 13, 17, 19\}

Explicit listing allows accurate evaluation of set operations.

Find the union

A \cup B

A \cup B = \{3, 4, 6, 8, 9, 12, 15, 16, 18, 20\}

The union combines all elements present in either

A

B

Determine the complement

(A \cup B)'

relative to

\mathcal{E}

(A \cup B)' = \{1, 2, 5, 7, 10, 11, 13, 14, 17, 19\}

The complement contains all elements of the universal set

\mathcal{E}

that are not in

A \cup B

Find the intersection

(A \cup B)' \cap C

and count its cardinality.

(A \cup B)' \cap C = \{2, 5, 7, 11, 13, 17, 19\}

, which contains 7 elements.

The intersection yields elements common to both

(A \cup B)'

and prime set

C

Set Complement and Intersections

Question