Questions: Assignment: 5.5 Sets
Score: 3 / 8 Answered: 3 / 8
Question 4
Let S be the universal set, where:
S=1,2,3, ..., 18,19,20
Let sets A and B be subsets of S, where:
Set A=2,5,6,8,10,12,13,16,19,20
Set B=1,2,5,7,9,10,12,14,15,16,19,20
Find the following:
LIST the elements in the set ( A ∩ Bc ):
(A ∩ Bc)=
Enter the elements as a list, separated by commas. If the result is the empty set, enter DNE LIST the elements in the set ( B ∩ Ac ):
(B ∩ Ac)=
Enter the elements as a list, separated by commas. If the result is the empty set, enter DNE You may want to draw a Venn Diagram to help answer this question.
Transcript text: Assignment: 5.5 Sets
Score: $3 / 8$ Answered: $3 / 8$
Question 4
Let $S$ be the universal set, where:
\[
S=\{1,2,3, \ldots, 18,19,20\}
\]
Let sets $A$ and $B$ be subsets of $S$, where:
Set $A=\{2,5,6,8,10,12,13,16,19,20\}$
Set $B=\{1,2,5,7,9,10,12,14,15,16,19,20\}$
Find the following:
LIST the elements in the set ( $A \cap B^{c}$ ):
\[
\left(A \cap B^{\subseteq}\right)=\{
\]
$\qquad$ \}
Enter the elements as a list, separated by commas. If the result is the empty set, enter DNE LIST the elements in the set ( $B \cap A^{\mathrm{c}}$ ):
$\left(B \cap A^{C}\right)=\{$ $\square$
Enter the elements as a list, separated by commas. If the result is the empty set, enter DNE You may want to draw a Venn Diagram to help answer this question.
Solution
Solution Steps
Step 1: Define the Universal Set $S$
The universal set $S$ is defined by the range from 1 to 20, inclusive.
So, $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}$.
Step 2: Calculate the Union of $A \cup B$
The union of $A$ and $B$, denoted as $A \cup B$, combines all unique elements from both sets.
So, $A \cup B = \{1, 2, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 19, 20\}$.
Step 3: Calculate the Intersection of $A \cap B$
The intersection of $A$ and $B$, denoted as $A \cap B$, lists all elements present in both sets.
So, $A \cap B = \{2, 5, 10, 12, 16, 19, 20\}$.
Step 4: Calculate the Complements $A^c$, $B^c$, and $C^c$
The complement of $A$ within the universal set $S$, denoted as $A^c$, lists all elements of $S$ not in $A$.
So, $A^c = \{1, 3, 4, 7, 9, 11, 14, 15, 17, 18\}$.
The complement of $B$ within the universal set $S$, denoted as $B^c$, lists all elements of $S$ not in $B$.
So, $B^c = \{3, 4, 6, 8, 11, 13, 17, 18\}$.
No set $C$ was provided, so $C^c$ is not applicable.
Final Answer:
The union $A \cup B = \{1, 2, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 19, 20\}$,
the intersection $A \cap B = \{2, 5, 10, 12, 16, 19, 20\}$,
the complement $A^c = \{1, 3, 4, 7, 9, 11, 14, 15, 17, 18\}$,
and $B^c = \{3, 4, 6, 8, 11, 13, 17, 18\}$.