Questions: Sets (B) and (C) are subsets of the universal set (U). These sets are defined as follows. [ U=1,3,4,5,6,7 B=3,4,6,7 C=1,4 ] Find the following sets. Write your answer in roster form or as (varnothing). (a) ((B cap C)^prime=) (square) (b) (B^prime cup C=) (square)

Sets (B) and (C) are subsets of the universal set (U).
These sets are defined as follows.
[
U=1,3,4,5,6,7
B=3,4,6,7
C=1,4
]

Find the following sets.
Write your answer in roster form or as (varnothing).
(a) ((B cap C)^prime=) (square)
(b) (B^prime cup C=) (square)

Transcript text: Sets $B$ and $C$ are subsets of the universal set $U$. These sets are defined as follows. \[ \begin{array}{l} U=\{1,3,4,5,6,7\} \\ B=\{3,4,6,7\} \\ C=\{1,4\} \] Find the following sets. Write your answer in roster form or as $\varnothing$. (a) $(B \cap C)^{\prime}=$ $\square$ (b) $B^{\prime} \cup C=$ $\square$

Solution

Solution Steps

To solve the given set problems, we need to understand the operations involved:

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

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

Step 1: Find $ B \cap C $

To find the intersection of sets $ B $ and $ C $, we calculate: \[ B \cap C = \{3, 4, 6, 7\} \cap \{1, 4\} = \{4\} \]

Step 2: Find $ (B \cap C)^{\prime} $

Next, we find the complement of the intersection with respect to the universal set $ U $: \[ (B \cap C)^{\prime} = U - (B \cap C) = \{1, 3, 4, 5, 6, 7\} - \{4\} = \{1, 3, 5, 6, 7\} \]

Step 3: Find $ B^{\prime} $

Now, we calculate the complement of set $ B $ with respect to the universal set $ U $: \[ B^{\prime} = U - B = \{1, 3, 4, 5, 6, 7\} - \{3, 4, 6, 7\} = \{1, 5\} \]

Step 4: Find $ B^{\prime} \cup C $

Finally, we find the union of $ B^{\prime} $ and $ C $: \[ B^{\prime} \cup C = \{1, 5\} \cup \{1, 4\} = \{1, 4, 5\} \]