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3. Derive AC ∪ (B ∩ C) when A = {x : 0 ≤ x ≤ 1}, B = {x : 0 ≤ x ≤ 3}, and C = {...

Jun 12, 2024

3. Derive AC ∪ (B ∩ C) when A = {x : 0 ≤ x ≤ 1}, B = {x : 0 ≤ x ≤ 3}, and C = {x : −1 ≤ x ≤ 2}. 4. Derive A ∩ B ∩ CC when A = {x : 0 ≤ x ≤ 1}, B = {x : 0 ≤ x ≤ 3}, and C = {x : −1 ≤ x ≤ 2}. 5. Derive [(A ∪ B) ∩ CC ]C when A = {x : 0 ≤ x ≤ 1}, B = {x : 0 ≤ x ≤ 3}, and C = {x : −1 ≤ x ≤ 2}

Solution by Steps

step 1

Determine

A^C

: A^C = \{x : x < 0 \text{ or } x > 1\}

step 2

Determine

B \cap C

B \cap C = \{x : 0 \leq x \leq 2\}

step 3

Determine

A^C \cup (B \cap C)

: A^C \cup (B \cap C) = \{x : x < 0 \text{ or } 0 \leq x \leq 2 \text{ or } x > 1\} = \{x : x < 0 \text{ or } x \leq 2\}

3 Answer

A^C \cup (B \cap C) = \{x : x < 0 \text{ or } x \leq 2\}

Key Concept

Union and Intersection of Sets

Explanation

The union of the complement of set

A

and the intersection of sets

B

and

C

is derived by combining the elements that are either not in

A

or in both

B

and

C

Question 4: Derive

A \cap B \cap C^C

when

A = \{x : 0 \leq x \leq 1\}

B = \{x : 0 \leq x \leq 3\}

, and

C = \{x : -1 \leq x \leq 2\}

step 1

Determine

C^C

: C^C = \{x : x < -1 \text{ or } x > 2\}

step 2

Determine

A \cap B

A \cap B = \{x : 0 \leq x \leq 1\}

step 3

Determine

A \cap B \cap C^C

: A \cap B \cap C^C = \{x : 0 \leq x \leq 1\} \cap \{x : x < -1 \text{ or } x > 2\} = \emptyset

4 Answer

A \cap B \cap C^C = \emptyset

Key Concept

Intersection of Sets and Complement

Explanation

The intersection of sets

A

and

B

with the complement of set

C

results in an empty set because there are no elements that satisfy all conditions simultaneously.

Question 5: Derive

[(A \cup B) \cap C^C]^C

when

A = \{x : 0 \leq x \leq 1\}

B = \{x : 0 \leq x \leq 3\}

, and

C = \{x : -1 \leq x \leq 2\}

step 1

Determine

A \cup B

A \cup B = \{x : 0 \leq x \leq 3\}

step 2

Determine

C^C

: C^C = \{x : x < -1 \text{ or } x > 2\}

step 3

Determine

(A \cup B) \cap C^C

: (A \cup B) \cap C^C = \{x : 0 \leq x \leq 3\} \cap \{x : x < -1 \text{ or } x > 2\} = \{x : 2 < x \leq 3\}

step 4

Determine

[(A \cup B) \cap C^C]^C

: [(A \cup B) \cap C^C]^C = \{x : x \leq 2 \text{ or } x > 3\}

5 Answer

[(A \cup B) \cap C^C]^C = \{x : x \leq 2 \text{ or } x > 3\}

Key Concept

Complement of Intersection and Union of Sets

Explanation

The complement of the intersection of the union of sets

A

and

B

with the complement of set

C

is derived by finding the elements that do not satisfy the intersection condition.