Question

let u be the universal set, where:
u = {1,2,3,…,18,19,20}
let sets a and b be subsets of u, where:
set a = {4,5,7,13,14,15,16,20}
set b = {4,12,13,15,17,20}
find the following:
list the elements in the set ( a cup b ):
( a cup b = )
enter the elements as a list, separated by commas. if the result is the empty set, enter dne
list the elements in the set ( a cap b ):
( a cap b = )
enter the elements as a list, separated by commas. if the result is the empty set, enter dne
you may want to draw a venn diagram to help answer this question.

Explanation:

For \( A \cup B \):

Step1: Recall union definition

The union of two sets \( A \) and \( B \), denoted \( A \cup B \), is the set of all elements that are in \( A \) or in \( B \) (or in both).

Step2: List elements of \( A \) and \( B \)

Set \( A = \{4, 5, 7, 13, 14, 15, 16, 20\} \), Set \( B = \{4, 12, 13, 15, 17, 20\} \).

Step3: Combine and remove duplicates

Combine the elements: \( 4, 5, 7, 13, 14, 15, 16, 20, 4, 12, 13, 15, 17, 20 \). Remove duplicates to get \( \{4, 5, 7, 12, 13, 14, 15, 16, 17, 20\} \).

For \( A \cap B \):

Step1: Recall intersection definition

The intersection of two sets \( A \) and \( B \), denoted \( A \cap B \), is the set of all elements that are in both \( A \) and \( B \).

Step2: Find common elements

Check which elements are in both \( A \) and \( B \). From \( A = \{4, 5, 7, 13, 14, 15, 16, 20\} \) and \( B = \{4, 12, 13, 15, 17, 20\} \), the common elements are \( 4, 13, 15, 20 \).

Answer:

s:
\( A \cup B = \{4, 5, 7, 12, 13, 14, 15, 16, 17, 20\} \)
\( A \cap B = \{4, 13, 15, 20\} \)