Math / Discrete

Question2. Let $A=\{x \in \mathbb{Z} \mid x$ is even $\}, B=\{5,7,8,12,13,15\}$ , and $C=\{3,5,9,12,15,16\}$ . The universal set $U$ is the set of all integers. Which of the following set operations returns the set $\{3,9\}$ ? a. $(A \cup B) \cap C$ c. $C-(A \cap B)$ b. $(\overline{A \cup B}) \cap C$ d. $C-(\overline{A \cap B})$

Studdy Solution

STEP 1

1. We are given three sets $A$ , $B$ , and $C$ with specific elements or properties.
2. The universal set $U$ is the set of all integers.
3. We need to determine which set operation results in the set $\{3,9\}$ .

STEP 2

1. Identify the elements in each set.
2. Evaluate each set operation.
3. Compare the result of each operation to $\{3,9\}$ .
4. Identify the correct set operation.

STEP 3

Identify the elements in each set: - Set $A$ is the set of all even integers. - Set $B = \{5, 7, 8, 12, 13, 15\}$ . - Set $C = \{3, 5, 9, 12, 15, 16\}$ .

STEP 4

Evaluate each set operation:
a. $(A \cup B) \cap C$ : - $A \cup B$ includes all even integers and the elements of $B$ . - $(A \cup B) \cap C$ is the intersection of $A \cup B$ and $C$ .

STEP 5

Calculate $(A \cup B) \cap C$ : - $A \cup B = \{\text{all even integers}\} \cup \{5, 7, 8, 12, 13, 15\}$ . - The intersection with $C = \{3, 5, 9, 12, 15, 16\}$ results in $\{5, 12, 15\}$ .

STEP 6

b. $(\overline{A \cup B}) \cap C$ : - $\overline{A \cup B}$ is the complement of $A \cup B$ , meaning all integers not in $A \cup B$ . - $(\overline{A \cup B}) \cap C$ is the intersection of $\overline{A \cup B}$ and $C$ .

STEP 7

Calculate $(\overline{A \cup B}) \cap C$ : - Since $A \cup B$ includes all even integers and the elements of $B$ , $\overline{A \cup B}$ includes all odd integers not in $B$ . - The intersection with $C = \{3, 5, 9, 12, 15, 16\}$ results in $\{3, 9\}$ .

STEP 8

c. $C - (A \cap B)$ : - $A \cap B$ is the intersection of $A$ and $B$ , which includes only even numbers from $B$ . - $C - (A \cap B)$ is the set difference, removing elements of $A \cap B$ from $C$ .

STEP 9

Calculate $C - (A \cap B)$ : - $A \cap B = \{8, 12\}$ (even numbers in $B$ ). - Removing these from $C = \{3, 5, 9, 12, 15, 16\}$ results in $\{3, 5, 9, 15, 16\}$ .

STEP 10

d. $C - (\overline{A \cap B})$ : - $\overline{A \cap B}$ is the complement of $A \cap B$ , meaning all integers not in $A \cap B$ . - $C - (\overline{A \cap B})$ is the set difference, removing elements of $\overline{A \cap B}$ from $C$ .

STEP 11

Calculate $C - (\overline{A \cap B})$ : - Since $A \cap B = \{8, 12\}$ , $\overline{A \cap B}$ includes all integers except $8$ and $12$ . - Removing these from $C = \{3, 5, 9, 12, 15, 16\}$ results in an empty set, as no elements are removed.

STEP 12

Compare the results to $\{3,9\}$ : - Option b, $(\overline{A \cup B}) \cap C$ , results in $\{3, 9\}$ .
The correct set operation is option b: $(\overline{A \cup B}) \cap C$ .

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Studdy Solution

STEP 1

1. We are given three sets $A$ , $B$ , and $C$ with specific elements or properties.
2. The universal set $U$ is the set of all integers.
3. We need to determine which set operation results in the set $\{3,9\}$ .

STEP 2

1. Identify the elements in each set.
2. Evaluate each set operation.
3. Compare the result of each operation to $\{3,9\}$ .
4. Identify the correct set operation.

STEP 3

Identify the elements in each set: - Set $A$ is the set of all even integers. - Set $B = \{5, 7, 8, 12, 13, 15\}$ . - Set $C = \{3, 5, 9, 12, 15, 16\}$ .

STEP 4

Evaluate each set operation:
a. $(A \cup B) \cap C$ : - $A \cup B$ includes all even integers and the elements of $B$ . - $(A \cup B) \cap C$ is the intersection of $A \cup B$ and $C$ .

STEP 5

Calculate $(A \cup B) \cap C$ : - $A \cup B = \{\text{all even integers}\} \cup \{5, 7, 8, 12, 13, 15\}$ . - The intersection with $C = \{3, 5, 9, 12, 15, 16\}$ results in $\{5, 12, 15\}$ .

STEP 6

b. $(\overline{A \cup B}) \cap C$ : - $\overline{A \cup B}$ is the complement of $A \cup B$ , meaning all integers not in $A \cup B$ . - $(\overline{A \cup B}) \cap C$ is the intersection of $\overline{A \cup B}$ and $C$ .

STEP 7

Calculate $(\overline{A \cup B}) \cap C$ : - Since $A \cup B$ includes all even integers and the elements of $B$ , $\overline{A \cup B}$ includes all odd integers not in $B$ . - The intersection with $C = \{3, 5, 9, 12, 15, 16\}$ results in $\{3, 9\}$ .

STEP 8

c. $C - (A \cap B)$ : - $A \cap B$ is the intersection of $A$ and $B$ , which includes only even numbers from $B$ . - $C - (A \cap B)$ is the set difference, removing elements of $A \cap B$ from $C$ .

STEP 9

Calculate $C - (A \cap B)$ : - $A \cap B = \{8, 12\}$ (even numbers in $B$ ). - Removing these from $C = \{3, 5, 9, 12, 15, 16\}$ results in $\{3, 5, 9, 15, 16\}$ .

STEP 10

d. $C - (\overline{A \cap B})$ : - $\overline{A \cap B}$ is the complement of $A \cap B$ , meaning all integers not in $A \cap B$ . - $C - (\overline{A \cap B})$ is the set difference, removing elements of $\overline{A \cap B}$ from $C$ .

STEP 11

Calculate $C - (\overline{A \cap B})$ : - Since $A \cap B = \{8, 12\}$ , $\overline{A \cap B}$ includes all integers except $8$ and $12$ . - Removing these from $C = \{3, 5, 9, 12, 15, 16\}$ results in an empty set, as no elements are removed.

STEP 12

Compare the results to $\{3,9\}$ : - Option b, $(\overline{A \cup B}) \cap C$ , results in $\{3, 9\}$ .
The correct set operation is option b: $(\overline{A \cup B}) \cap C$ .

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Studdy Solution

STEP 1

1. We are given three sets A A A, B B B, and C C C with specific elements or properties.2. The universal set U U U is the set of all integers.3. We need to determine which set operation results in the set {3,9}\{3,9\}{3,9}.

STEP 2

1. Identify the elements in each set.2. Evaluate each set operation.3. Compare the result of each operation to {3,9}\{3,9\}{3,9}.4. Identify the correct set operation.

STEP 3

Identify the elements in each set: - Set A A A is the set of all even integers. - Set B={5,7,8,12,13,15} B = \{5, 7, 8, 12, 13, 15\} B={5,7,8,12,13,15}. - Set C={3,5,9,12,15,16} C = \{3, 5, 9, 12, 15, 16\} C={3,5,9,12,15,16}.

STEP 4

Evaluate each set operation:a. (A∪B)∩C(A \cup B) \cap C(A∪B)∩C: - A∪B A \cup B A∪B includes all even integers and the elements of B B B. - (A∪B)∩C(A \cup B) \cap C(A∪B)∩C is the intersection of A∪B A \cup B A∪B and C C C.

STEP 5

STEP 6

STEP 7

STEP 8

c. C−(A∩B) C - (A \cap B) C−(A∩B): - A∩B A \cap B A∩B is the intersection of A A A and B B B, which includes only even numbers from B B B. - C−(A∩B) C - (A \cap B) C−(A∩B) is the set difference, removing elements of A∩B A \cap B A∩B from C C C.

STEP 9

Calculate C−(A∩B) C - (A \cap B) C−(A∩B): - A∩B={8,12} A \cap B = \{8, 12\} A∩B={8,12} (even numbers in B B B). - Removing these from C={3,5,9,12,15,16} C = \{3, 5, 9, 12, 15, 16\}C={3,5,9,12,15,16} results in {3,5,9,15,16}\{3, 5, 9, 15, 16\}{3,5,9,15,16}.

STEP 10

STEP 11

STEP 12

Compare the results to {3,9}\{3,9\}{3,9}: - Option b, (A∪B‾)∩C(\overline{A \cup B}) \cap C(A∪B)∩C, results in {3,9}\{3, 9\}{3,9}. The correct set operation is option b: (A∪B‾)∩C(\overline{A \cup B}) \cap C(A∪B)∩C.

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1. We are given three sets $A$ , $B$ , and $C$ with specific elements or properties.
2. The universal set $U$ is the set of all integers.
3. We need to determine which set operation results in the set $\{3,9\}$ .

1. Identify the elements in each set.
2. Evaluate each set operation.
3. Compare the result of each operation to $\{3,9\}$ .
4. Identify the correct set operation.

Identify the elements in each set: - Set $A$ is the set of all even integers. - Set $B = \{5, 7, 8, 12, 13, 15\}$ . - Set $C = \{3, 5, 9, 12, 15, 16\}$ .

Evaluate each set operation:
a. $(A \cup B) \cap C$ : - $A \cup B$ includes all even integers and the elements of $B$ . - $(A \cup B) \cap C$ is the intersection of $A \cup B$ and $C$ .

c. $C - (A \cap B)$ : - $A \cap B$ is the intersection of $A$ and $B$ , which includes only even numbers from $B$ . - $C - (A \cap B)$ is the set difference, removing elements of $A \cap B$ from $C$ .

Calculate $C - (A \cap B)$ : - $A \cap B = \{8, 12\}$ (even numbers in $B$ ). - Removing these from $C = \{3, 5, 9, 12, 15, 16\}$ results in $\{3, 5, 9, 15, 16\}$ .

Compare the results to $\{3,9\}$ : - Option b, $(\overline{A \cup B}) \cap C$ , results in $\{3, 9\}$ .
The correct set operation is option b: $(\overline{A \cup B}) \cap C$ .