Questions: These sets are defined as U=f, k, m, q, r, x, z B=k, r, z D=f, m, r, z Find the following sets. Write your answer in roster form or as ∅. (a) B ∩ D′= (b) (B ∪ D)′=

Transcript text: These sets are defined as \[ \begin{array}{l} U=\{f, k, m, q, r, x, z\} \\ B=\{k, r, z\} \\ D=\{f, m, r, z\} \end{array} \] Find the following sets. Write your answer in roster form or as $\varnothing$. (a) $B \cap D^{\prime}=$ (b) $(B \cup D)^{\prime}=$

Solution

Solution Steps

To solve the given set problems, we need to perform set operations such as intersection, union, and complement.

(a) To find $ B \cap D^{\prime} $, we first determine the complement of set $ D $ with respect to the universal set $ U $, and then find the intersection of set $ B $ with this complement.

(b) To find $ (B \cup D)^{\prime} $, we first find the union of sets $ B $ and $ D $, and then determine the complement of this union with respect to the universal set $ U $.

Step 1: Find $ D^{\prime} $

To find the complement of set $ D $ with respect to the universal set $ U $, we calculate: \[ D^{\prime} = U - D = \{f, k, m, q, r, x, z\} - \{f, m, r, z\} = \{q, k, x\} \]

Step 2: Calculate $ B \cap D^{\prime} $

Next, we find the intersection of set $ B $ and the complement of $ D $: \[ B \cap D^{\prime} = \{k, r, z\} \cap \{q, k, x\} = \{k\} \]

Step 3: Find $ B \cup D $

We then calculate the union of sets $ B $ and $ D $: \[ B \cup D = \{k, r, z\} \cup \{f, m, r, z\} = \{f, k, m, r, z\} \]

Step 4: Calculate $ (B \cup D)^{\prime} $

Finally, we find the complement of the union $ B \cup D $ with respect to $ U $: \[ (B \cup D)^{\prime} = U - (B \cup D) = \{f, k, m, q, r, x, z\} - \{f, k, m, r, z\} = \{q, x\} \]