Questions: Set C and the universal set U are defined as follows. U=1,2,3,4,5,6,7 C=2,4,7 Find the following sets. Write your answer in roster form or as varnothing. (a) C^prime cap U= square (b) C cup varnothing= square

Set C and the universal set U are defined as follows.
U=1,2,3,4,5,6,7
C=2,4,7

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

Transcript text: Set $C$ and the universal set $U$ are defined as follows. \[ \begin{array}{l} U=\{1,2,3,4,5,6,7\} \\ C=\{2,4,7\} \end{array} \] Find the following sets. Write your answer in roster form or as $\varnothing$. (a) $C^{\prime} \cap U=$ $\square$ (b) $C \cup \varnothing=$ $\square$

Solution

Solution Steps

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

(a) The complement of set $ C $ with respect to the universal set $ U $ is the set of elements in $ U $ that are not in $ C $. The intersection of this complement with $ U $ will simply be the complement itself since it is already a subset of $ U $.

(b) The union of any set with the empty set $ \varnothing $ is the set itself.

Step 1: Define the Universal Set $ U $ and Set $ C $

Given: \[ U = \{1, 2, 3, 4, 5, 6, 7\} \] \[ C = \{2, 4, 7\} \]

Step 2: Find the Complement of $ C $ with Respect to $ U $

The complement of $ C $ with respect to $ U $ is the set of elements in $ U $ that are not in $ C $: \[ C' = U \setminus C = \{1, 3, 5, 6\} \]

Step 3: Find $ C' \cap U $

Since $ C' $ is already a subset of $ U $, the intersection of $ C' $ with $ U $ is simply $ C' $: \[ C' \cap U = \{1, 3, 5, 6\} \]

Step 4: Find $ C \cup \varnothing $

The union of any set with the empty set $ \varnothing $ is the set itself: \[ C \cup \varnothing = C = \{2, 4, 7\} \]