Answered

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Let [tex]$U = \{\text{all integers}\}$[/tex].

Consider the following sets:

[tex]A=\{x \mid x \in U \text{ and } x\ \textgreater \ 3\}[/tex]
[tex]B=\{x \mid x \in U \text{ and } x \text{ is an even integer}\}[/tex]
[tex]C=\{x \mid x \in U \text{ and } 2x \text{ is an odd integer}\}[/tex]
[tex]D =\{x \mid x \in U \text{ and } x \text{ is an odd integer}\}[/tex]

Use the defined sets to answer the questions.

1. Assuming 0 is an even integer, which set is the complement to set [tex]$B$[/tex]?
[tex]\square[/tex]

2. Which set is an empty set?
[tex]\square[/tex]

3. Which set would contain the subset [tex]$E=\{1, 3, 5, 7\}$[/tex]?
[tex]\square[/tex]

Sagot :

GinnyAnswer
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]
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