Let’s address each part of the question step by step.
### 1. Which set is the complement to set [tex]\( B \)[/tex]?
Set [tex]\( B \)[/tex] contains all even integers. The complement of set [tex]\( B \)[/tex] ([tex]\(\overline{B}\)[/tex]) consists of all elements in the universe [tex]\( U \)[/tex] (all integers) that are not in [tex]\( B \)[/tex]. These would be all the odd integers. Therefore:
[tex]\[ \overline{B} = \{x \mid x \in U \text{ and } x \text{ is odd} \} \][/tex]
Hence, the complement to set [tex]\( B \)[/tex] is:
[tex]\[ \overline{B} = \{1, 3, 5, 7, \ldots, -3, -1, -997, -999, \text{etc.}\} \][/tex]
### 2. Which set is an empty set?
To identify which set is empty, we examine the definitions of the sets given:
- [tex]\(A\)[/tex] contains integers greater than 3.
- [tex]\(B\)[/tex] contains even integers.
- [tex]\(C\)[/tex] contains integers [tex]\( x \)[/tex] such that [tex]\( 2x \)[/tex] is odd. Notice that [tex]\( 2x \)[/tex] is always even for any integer [tex]\( x \)[/tex], hence [tex]\( 2x \)[/tex] cannot be odd. This implies that there is no such [tex]\( x \)[/tex], making set [tex]\( C \)[/tex] empty.
- [tex]\(D\)[/tex] contains odd integers.
Therefore, set [tex]\( C \)[/tex] is:
[tex]\[ C = \emptyset \][/tex]
### 3. Which set would contain the subset [tex]\( E = \{1, 3, 5, 7\} \)[/tex]?
Subset [tex]\( E \)[/tex] includes the integers 1, 3, 5, and 7. These integers are all odd. Therefore, we look for a set among [tex]\( A, B, C, D \)[/tex] that contains all odd integers. We can see:
- [tex]\( A \)[/tex] contains integers greater than 3, which does include some of the elements in [tex]\( E \)[/tex] (namely, 5 and 7), but not all of them.
- [tex]\( B \)[/tex] contains even integers, which does not include any element of [tex]\( E \)[/tex].
- [tex]\( C \)[/tex] is empty, hence it does not contain [tex]\( E \)[/tex].
- [tex]\( D \)[/tex] contains all odd integers, which would include all elements of [tex]\( E \)[/tex].
Thus, the set [tex]\( D \)[/tex] contains the subset [tex]\( E \)[/tex]:
[tex]\[ D \supseteq \{1, 3, 5, 7\} \][/tex]
### Summary
1. The complement to set [tex]\( B \)[/tex] is:
[tex]\[ \{1, 3, 5, 7, \ldots, -3, -1, -997, -999, \text{etc.}\} \][/tex]
2. The empty set is:
[tex]\[ C = \emptyset \][/tex]
3. The set containing the subset [tex]\( E = \{1, 3, 5, 7\} \)[/tex] is:
[tex]\[ D \][/tex]
