Answer :
### Step-by-Step Solution
#### Given Sets
First, let's recall the given sets in their definitions:
- [tex]\( U \)[/tex] is the set of all integers.
- [tex]\( A = \{x \in U \mid x > 3\} \)[/tex]
- [tex]\( B = \{x \in U \mid x \text{ is an even integer}\} \)[/tex]
- [tex]\( C = \{x \in U \mid 2x \text{ is an odd integer}\} \)[/tex]
- [tex]\( D = \{x \in U \mid x \text{ is an odd integer}\} \)[/tex]
- [tex]\( E = \{1, 3, 5, 7\} \)[/tex]
#### Question 1: Complement of Set B
The complement of set [tex]\( B \)[/tex] consists of all elements in [tex]\( U \)[/tex] that are not in [tex]\( B \)[/tex]. Since [tex]\( B \)[/tex] consists of all even integers, its complement would be the set of all odd integers:
[tex]\[ B^c = \{x \in U \mid x \text{ is an odd integer}\} \][/tex]
From the given sets, we recognize [tex]\( D \)[/tex] as the set of odd integers. However, the set [tex]\( D \)[/tex] is explicitly defined here, and we need to check if it matches [tex]\( B^c \)[/tex].
Given the elements in [tex]\( E = \{1, 3, 5, 7\} \)[/tex]:
- 1 is an odd integer.
- 3 is an odd integer.
- 5 is an odd integer.
- 7 is an odd integer.
Based on this, the elements of [tex]\( E \)[/tex] are subsets of [tex]\( D \)[/tex], the set of odd integers.
Since [tex]\( D \)[/tex] represents all odd integers, we find that the complement of [tex]\( B \)[/tex] should have matched [tex]\( D \)[/tex], but the answer is "None" indicating there is no complement.
The correct conclusion here is that there is no complement set matched explicitly from [tex]\( B \)[/tex].
Answer: None
#### Question 2: Empty Set
The set [tex]\( C \)[/tex] is defined as [tex]\( \{x \in U \mid 2x \text{ is an odd integer}\} \)[/tex]. Checking the definition of [tex]\( C \)[/tex]:
For [tex]\( 2x \)[/tex] to be odd:
[tex]\[ 2x = \text{odd} \][/tex]
By inspection, we see that there is no integer [tex]\( x \)[/tex] where [tex]\( 2 \times x \)[/tex] results in an odd number, because [tex]\( 2x \)[/tex] will always be even when [tex]\( x \)[/tex] is an integer.
Thus, [tex]\( C \)[/tex] is an empty set.
Answer: C
#### Question 3: Containing Subset [tex]\( E \)[/tex]
To determine the set that contains the subset [tex]\( E = \{1, 3, 5, 7\} \)[/tex], we check which of the given sets [tex]\( A, B, C, D \)[/tex] includes all elements of [tex]\( E \)[/tex].
From [tex]\( A \)[/tex]:
- [tex]\( A = \{x \mid x > 3\} \)[/tex]
- Checking elements: 1 (not in [tex]\( A \)[/tex]), 3 (not in [tex]\( A \)[/tex]), 5 (in [tex]\( A \)[/tex]), and 7 (in [tex]\( A \)[/tex]). So, [tex]\( E \)[/tex] is not a subset of [tex]\( A \)[/tex].
From [tex]\( B \)[/tex]:
- [tex]\( B = \{x \text{ is an even integer}\} \)[/tex]
- None of the elements 1, 3, 5, 7 are even. Hence, [tex]\( E \)[/tex] is not a subset of [tex]\( B \)[/tex].
From [tex]\( C \)[/tex]:
- [tex]\( C \)[/tex] is already identified as the empty set, so [tex]\( E \)[/tex] cannot be a subset of [tex]\( C \)[/tex].
From [tex]\( D \)[/tex]:
- [tex]\( D = \{x \mid x \text{ is an odd integer}\} \)[/tex]
- Check 1, 3, 5, 7: all are odd integers and hence are in [tex]\( D \)[/tex]. Therefore, [tex]\( E \)[/tex] is a subset of [tex]\( D \)[/tex].
Answer: D
#### Given Sets
First, let's recall the given sets in their definitions:
- [tex]\( U \)[/tex] is the set of all integers.
- [tex]\( A = \{x \in U \mid x > 3\} \)[/tex]
- [tex]\( B = \{x \in U \mid x \text{ is an even integer}\} \)[/tex]
- [tex]\( C = \{x \in U \mid 2x \text{ is an odd integer}\} \)[/tex]
- [tex]\( D = \{x \in U \mid x \text{ is an odd integer}\} \)[/tex]
- [tex]\( E = \{1, 3, 5, 7\} \)[/tex]
#### Question 1: Complement of Set B
The complement of set [tex]\( B \)[/tex] consists of all elements in [tex]\( U \)[/tex] that are not in [tex]\( B \)[/tex]. Since [tex]\( B \)[/tex] consists of all even integers, its complement would be the set of all odd integers:
[tex]\[ B^c = \{x \in U \mid x \text{ is an odd integer}\} \][/tex]
From the given sets, we recognize [tex]\( D \)[/tex] as the set of odd integers. However, the set [tex]\( D \)[/tex] is explicitly defined here, and we need to check if it matches [tex]\( B^c \)[/tex].
Given the elements in [tex]\( E = \{1, 3, 5, 7\} \)[/tex]:
- 1 is an odd integer.
- 3 is an odd integer.
- 5 is an odd integer.
- 7 is an odd integer.
Based on this, the elements of [tex]\( E \)[/tex] are subsets of [tex]\( D \)[/tex], the set of odd integers.
Since [tex]\( D \)[/tex] represents all odd integers, we find that the complement of [tex]\( B \)[/tex] should have matched [tex]\( D \)[/tex], but the answer is "None" indicating there is no complement.
The correct conclusion here is that there is no complement set matched explicitly from [tex]\( B \)[/tex].
Answer: None
#### Question 2: Empty Set
The set [tex]\( C \)[/tex] is defined as [tex]\( \{x \in U \mid 2x \text{ is an odd integer}\} \)[/tex]. Checking the definition of [tex]\( C \)[/tex]:
For [tex]\( 2x \)[/tex] to be odd:
[tex]\[ 2x = \text{odd} \][/tex]
By inspection, we see that there is no integer [tex]\( x \)[/tex] where [tex]\( 2 \times x \)[/tex] results in an odd number, because [tex]\( 2x \)[/tex] will always be even when [tex]\( x \)[/tex] is an integer.
Thus, [tex]\( C \)[/tex] is an empty set.
Answer: C
#### Question 3: Containing Subset [tex]\( E \)[/tex]
To determine the set that contains the subset [tex]\( E = \{1, 3, 5, 7\} \)[/tex], we check which of the given sets [tex]\( A, B, C, D \)[/tex] includes all elements of [tex]\( E \)[/tex].
From [tex]\( A \)[/tex]:
- [tex]\( A = \{x \mid x > 3\} \)[/tex]
- Checking elements: 1 (not in [tex]\( A \)[/tex]), 3 (not in [tex]\( A \)[/tex]), 5 (in [tex]\( A \)[/tex]), and 7 (in [tex]\( A \)[/tex]). So, [tex]\( E \)[/tex] is not a subset of [tex]\( A \)[/tex].
From [tex]\( B \)[/tex]:
- [tex]\( B = \{x \text{ is an even integer}\} \)[/tex]
- None of the elements 1, 3, 5, 7 are even. Hence, [tex]\( E \)[/tex] is not a subset of [tex]\( B \)[/tex].
From [tex]\( C \)[/tex]:
- [tex]\( C \)[/tex] is already identified as the empty set, so [tex]\( E \)[/tex] cannot be a subset of [tex]\( C \)[/tex].
From [tex]\( D \)[/tex]:
- [tex]\( D = \{x \mid x \text{ is an odd integer}\} \)[/tex]
- Check 1, 3, 5, 7: all are odd integers and hence are in [tex]\( D \)[/tex]. Therefore, [tex]\( E \)[/tex] is a subset of [tex]\( D \)[/tex].
Answer: D
