Let's analyze each part separately and match them correctly.
1. [tex]\(D \cap E\)[/tex]: This is the set of elements that are in both [tex]\(D\)[/tex] (whole numbers from 0 to 100) and [tex]\(E\)[/tex] (perfect squares less than 100).
[tex]\[
D \cap E = \{0, 1, 4, 9, 16, 25, 36, 49, 64, 81\}
\][/tex]
2. [tex]\(D \cap F\)[/tex]: This is the set of elements that are in both [tex]\(D\)[/tex] (whole numbers from 0 to 100) and [tex]\(F\)[/tex] (even numbers between 10 and 20).
[tex]\[
D \cap F = \{10, 12, 14, 16, 18\}
\][/tex]
3. [tex]\(D \cap (E \cap F)\)[/tex]: First, we find [tex]\(E \cap F\)[/tex], which is the set of elements that are both perfect squares less than 100 and even numbers between 10 and 20.
- [tex]\(E \cap F = \{16\}\)[/tex]
Then we take the intersection of this set with [tex]\(D\)[/tex]:
[tex]\[
D \cap (E \cap F) = \{16\}
\][/tex]
4. [tex]\(D \cup (E \cap F)\)[/tex]: Using the same [tex]\(E \cap F\)[/tex] from above:
[tex]\[
D \cup (E \cap F) = \text{All whole numbers from 0 to 100} \cup \{16\} = \text{All whole numbers from 0 to 100}
\][/tex]
5. [tex]\(D \cap (E \cup F)\)[/tex]: First, we find [tex]\(E \cup F\)[/tex], which is the set of elements that are either perfect squares less than 100 or even numbers between 10 and 20:
- [tex]\(E \cup F = \{0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 12, 14, 18\}\)[/tex]
Then we take the intersection of this set with [tex]\(D\)[/tex]:
[tex]\[
D \cap (E \cup F) = \{0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 12, 14, 18\}
\][/tex]
Matching the answers to their respective options:
1. [tex]\(D \cap E\)[/tex] matches [tex]\(\{0, 1, 4, 9, 16, 25, 36, 49, 64, 81\}\)[/tex].
2. [tex]\(D \cap F\)[/tex] matches [tex]\(\{10, 12, 14, 16, 18\}\)[/tex].
3. [tex]\(D \cap (E \cap F)\)[/tex] matches [tex]\(16\)[/tex].
4. [tex]\(D \cup (E \cap F)\)[/tex] matches "all whole numbers".
5. [tex]\(D \cap (E \cup F)\)[/tex] matches [tex]\(\{0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 12, 14, 18\}\)[/tex].
Therefore, the correctly completed matching is:
1. [tex]\(D \cap E\)[/tex] [tex]\( \rightarrow \)[/tex] [tex]\(\{0, 1, 4, 9, 16, 25, 36, 49, 64, 81\}\)[/tex]
2. [tex]\(D \cap F\)[/tex] [tex]\( \rightarrow \)[/tex] [tex]\(\{10, 12, 14, 16, 18\}\)[/tex]
3. [tex]\(D \cap (E \cap F)\)[/tex] [tex]\( \rightarrow \)[/tex] [tex]\(\{16\}\)[/tex]
4. [tex]\(D \cup (E \cap F)\)[/tex] [tex]\( \rightarrow \)[/tex] [tex]\(\text{All whole numbers}\)[/tex]
5. [tex]\(D \cap (E \cup F)\)[/tex] [tex]\( \rightarrow \)[/tex] [tex]\(\{0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 12, 14, 18\}\)[/tex]
