Question

let u be the universal set, where u = {1,2,3,4,5,6,7,8,9,10,11,12}. let sets a, b, and c be subsets of u, where a = {1,4,5,8,10} b = {1,6,7,9,11} c = {1,4,5,6,8,11,12}. find the results of each set operation below. give the elements of the set as a list, separated by commas. if the result is the empty set, enter dne. a. a ∪ ∅ = { } b. b ∩ c = { } c. c ∪ b = { } d. (a ∩ c) ∩ b = { } question help: video

Explanation:

Part a: \( A' \cup \emptyset \)

Step1: Find \( A' \) (complement of A)

The universal set \( U = \{1,2,3,4,5,6,7,8,9,10,11,12\} \) and \( A = \{1,4,5,8,10\} \). The complement \( A' \) contains elements in \( U \) but not in \( A \). So \( A' = \{2,3,6,7,9,11,12\} \).

Step2: Union with empty set

The union of any set with the empty set \( \emptyset \) is the set itself. So \( A' \cup \emptyset = A' = \{2,3,6,7,9,11,12\} \).

Step1: Identify common elements

Set \( B = \{1,6,7,9,11\} \) and set \( C = \{1,4,5,6,8,11,12\} \). The intersection \( B \cap C \) contains elements common to both \( B \) and \( C \).
Looking at the elements, common elements are 1, 6, 11.

Step2: Form the intersection set

So \( B \cap C = \{1,6,11\} \).

Step1: Find \( C' \) (complement of C)

Universal set \( U = \{1,2,3,4,5,6,7,8,9,10,11,12\} \) and \( C = \{1,4,5,6,8,11,12\} \). \( C' \) has elements in \( U \) not in \( C \), so \( C' = \{2,3,7,9,10\} \).

Step2: Union with B

Set \( B = \{1,6,7,9,11\} \). The union \( C' \cup B \) contains all elements in \( C' \) or \( B \). Combining \( \{2,3,7,9,10\} \) and \( \{1,6,7,9,11\} \), we get \( \{1,2,3,6,7,9,10,11\} \).

Answer:

2,3,6,7,9,11,12

Part b: \( B \cap C \)