Sure, let's go through the steps to solve the given set operations problem.
### 1. Draw a Venn Diagram
To draw a Venn diagram representing the universal set [tex]\( U \)[/tex] and the sets [tex]\( D \)[/tex], [tex]\( E \)[/tex], and [tex]\( F \)[/tex], we need to place each element in the appropriate region where these sets intersect or do not intersect.
- Universal Set [tex]\( U \)[/tex]: \{2, 4, 6, 8, 10, 12, 14, 16, 18\}
- Set [tex]\( D \)[/tex]: \{2, 4, 6, 8, 10\}
- Set [tex]\( E \)[/tex]: \{2, 4, 14, 16, 18\}
- Set [tex]\( F \)[/tex]: \{4, 10, 12, 16\}
When drawing the Venn diagram:
- Place 4 in the intersection of all three sets [tex]\(D\)[/tex], [tex]\(E\)[/tex], and [tex]\(F\)[/tex].
- Place other elements considering their presence in the combinations of sets (D, E, F).
Given that creating a visual Venn diagram is outside the scope of text format, you can visualize three overlapping circles, one each for [tex]\(D\)[/tex], [tex]\(E\)[/tex], and [tex]\(F\)[/tex], with their elements appropriately placed in the intersections.
### 2. Find [tex]\( D - (E \cup F) \)[/tex] and [tex]\( (E \cup F) - D \)[/tex]
#### Step-by-Step Solution:
Step 1: Calculate [tex]\( E \cup F \)[/tex]
The union of sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex] is a set containing all unique elements that are in either [tex]\( E \)[/tex] or [tex]\( F \)[/tex] or in both.
[tex]\[
E \cup F = \{2, 4, 14, 16, 18\} \cup \{4, 10, 12, 16\} = \{2, 4, 10, 12, 14, 16, 18\}
\][/tex]
Step 2: Calculate [tex]\( D - (E \cup F) \)[/tex]
This is the set of elements that are in [tex]\( D \)[/tex] but not in [tex]\( (E \cup F) \)[/tex].
Given:
[tex]\[
D = \{2, 4, 6, 8, 10\}
\][/tex]
[tex]\[
E \cup F = \{2, 4, 10, 12, 14, 16, 18\}
\][/tex]
Now, find the difference:
[tex]\[
D - (E \cup F) = \{6, 8\}
\][/tex]
Step 3: Calculate [tex]\( (E \cup F) - D \)[/tex]
This is the set of elements that are in [tex]\( (E \cup F) \)[/tex] but not in [tex]\( D \)[/tex].
[tex]\[
E \cup F = \{2, 4, 10, 12, 14, 16, 18\}
\][/tex]
[tex]\[
D = \{2, 4, 6, 8, 10\}
\][/tex]
Now, find the difference:
[tex]\[
(E \cup F) - D = \{12, 14, 16, 18\}
\][/tex]
### Final Results:
[tex]\[
D - (E \cup F) = \{6, 8\}
\][/tex]
[tex]\[
(E \cup F) - D = \{12, 14, 16, 18\}
\][/tex]
Thus, the detailed step-by-step solution demonstrates that:
[tex]\[
D - (E \cup F) = \{6, 8\}
\][/tex]
[tex]\[
(E \cup F) - D = \{12, 14, 16, 18\}
\][/tex]
