To solve the questions, we need to find the respective sets involved and then interpret the expressions.
### Sets Definition:
1. Set [tex]\( D \)[/tex]: All whole numbers.
[tex]\( D = \{0, 1, 2, 3, \dots, 100\} \)[/tex]
2. Set [tex]\( E \)[/tex]: Perfect squares between 49 and 100.
[tex]\( E = \{49, 64, 81, 100\} \)[/tex]
3. Set [tex]\( F \)[/tex]: Even numbers between 10 and 20.
[tex]\( F = \{12, 14, 16, 18\} \)[/tex]
### Expressions:
#### [tex]\( D \cup F \)[/tex]
The union of [tex]\( D \)[/tex] and [tex]\( F \)[/tex], which includes all elements from both sets without duplicates.
[tex]\( D \cup F = \{0, 1, 2, 3, \dots, 100\} \)[/tex]
#### [tex]\( D \cap F \)[/tex]
The intersection of [tex]\( D \)[/tex] and [tex]\( F \)[/tex], which includes only elements that are present in both sets.
[tex]\( D \cap F = \{12, 14, 16, 18\} \)[/tex]
#### [tex]\( D \cap E \)[/tex]
The intersection of [tex]\( D \)[/tex] and [tex]\( E \)[/tex], which includes only elements that are present in both sets.
[tex]\( D \cap E = \{64, 81, 100\} \)[/tex]
#### [tex]\( E \cap F \)[/tex]
The intersection of [tex]\( E \)[/tex] and [tex]\( F \)[/tex], which includes only elements that are present in both sets.
[tex]\( E \cap F = \{\} \)[/tex] (empty set as there are no common elements)
#### [tex]\( D \cap (E \cup F) \)[/tex]
The union of [tex]\( E \)[/tex] and [tex]\( F \)[/tex], then taking the intersection with [tex]\( D \)[/tex].
- First, find [tex]\( E \cup F \)[/tex]:
[tex]\( E \cup F = \{49, 64, 81, 100, 12, 14, 16, 18\} \)[/tex]
- Now, find the intersection with [tex]\( D \)[/tex]:
[tex]\( D \cap (E \cup F) = \{12, 14, 16, 18, 64, 81, 100\} \)[/tex]
### Filling the Blanks:
- [tex]\( D \cup F \)[/tex] means the set of all whole numbers from 0 to 100.
- [tex]\( D \cap F \)[/tex] means [tex]\( \{12, 14, 16, 18\} \)[/tex].
- [tex]\( D \cap E \)[/tex] means [tex]\( \{64, 81, 100\} \)[/tex].
- [tex]\( E \cap F \)[/tex] means [tex]\( \emptyset \)[/tex] (empty set).
- [tex]\( D \cap (E \cup F) \)[/tex] means [tex]\( \{12, 14, 16, 18, 64, 81, 100\} \)[/tex].
These are the detailed steps and interpretations for each given expression!
