Answer :
Certainly! Let's address each part of the problem one step at a time.
### a) Is [tex]\( C \subset B \)[/tex]? Explain why or why not.
To determine if [tex]\( C \)[/tex] is a subset of [tex]\( B \)[/tex], we need to check whether every element in [tex]\( C \)[/tex] is also in [tex]\( B \)[/tex]. The sets are as follows:
- [tex]\( C = \{2, 4, 6, 8, 10, 12\} \)[/tex]
- [tex]\( B = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24\} \)[/tex]
Each element in [tex]\( C \)[/tex] is present in [tex]\( B \)[/tex]. Therefore, [tex]\( C \)[/tex] is a subset of [tex]\( B \)[/tex]:
[tex]\[ C \subseteq B \][/tex]
### c) What is [tex]\( A \cap B \)[/tex]?
The intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted [tex]\( A \cap B \)[/tex], includes all elements that are present in both sets.
- [tex]\( A = \{1, 2, 3, 4, 6, 8, 12, 24\} \)[/tex]
- [tex]\( B = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24\} \)[/tex]
The common elements are [tex]\( \{2, 4, 6, 8, 12, 24\} \)[/tex]:
[tex]\[ A \cap B = \{2, 4, 6, 8, 12, 24\} \][/tex]
### d) What is [tex]\( A \cap C \)[/tex]?
The intersection of sets [tex]\( A \)[/tex] and [tex]\( C \)[/tex], denoted [tex]\( A \cap C \)[/tex], includes all elements that are present in both sets.
- [tex]\( A = \{1, 2, 3, 4, 6, 8, 12, 24\} \)[/tex]
- [tex]\( C = \{2, 4, 6, 8, 10, 12\} \)[/tex]
The common elements are [tex]\( \{2, 4, 6, 8, 12\} \)[/tex]:
[tex]\[ A \cap C = \{2, 4, 6, 8, 12\} \][/tex]
### e) What is [tex]\( A \cup B \)[/tex]?
The union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted [tex]\( A \cup B \)[/tex], includes all elements that are present in either set or both sets.
- [tex]\( A = \{1, 2, 3, 4, 6, 8, 12, 24\} \)[/tex]
- [tex]\( B = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24\} \)[/tex]
The combined set is:
[tex]\[ A \cup B = \{1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24\} \][/tex]
### f) What is [tex]\( A \cup C \)[/tex]?
The union of sets [tex]\( A \)[/tex] and [tex]\( C \)[/tex], denoted [tex]\( A \cup C \)[/tex], includes all elements that are present in either set or both sets.
- [tex]\( A = \{1, 2, 3, 4, 6, 8, 12, 24\} \)[/tex]
- [tex]\( C = \{2, 4, 6, 8, 10, 12\} \)[/tex]
The combined set is:
[tex]\[ A \cup C = \{1, 2, 3, 4, 6, 8, 10, 12, 24\} \][/tex]
### g) What is [tex]\( A^c \)[/tex]?
The complement of set [tex]\( A \)[/tex] in the universal set [tex]\( U \)[/tex], denoted [tex]\( A^c \)[/tex], includes all elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex].
- [tex]\( A = \{1, 2, 3, 4, 6, 8, 12, 24\} \)[/tex]
- [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24\} \)[/tex]
The elements in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex] are:
[tex]\[ A^c = \{5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23\} \][/tex]
To summarize:
a) [tex]\( C \subseteq B \)[/tex] : True
c) [tex]\( A \cap B = \{2, 4, 6, 8, 12, 24\} \)[/tex]
d) [tex]\( A \cap C = \{2, 4, 6, 8, 12\} \)[/tex]
e) [tex]\( A \cup B = \{1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24\} \)[/tex]
f) [tex]\( A \cup C = \{1, 2, 3, 4, 6, 8, 10, 12, 24\} \)[/tex]
g) [tex]\( A^c = \{5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23\} \)[/tex]
### a) Is [tex]\( C \subset B \)[/tex]? Explain why or why not.
To determine if [tex]\( C \)[/tex] is a subset of [tex]\( B \)[/tex], we need to check whether every element in [tex]\( C \)[/tex] is also in [tex]\( B \)[/tex]. The sets are as follows:
- [tex]\( C = \{2, 4, 6, 8, 10, 12\} \)[/tex]
- [tex]\( B = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24\} \)[/tex]
Each element in [tex]\( C \)[/tex] is present in [tex]\( B \)[/tex]. Therefore, [tex]\( C \)[/tex] is a subset of [tex]\( B \)[/tex]:
[tex]\[ C \subseteq B \][/tex]
### c) What is [tex]\( A \cap B \)[/tex]?
The intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted [tex]\( A \cap B \)[/tex], includes all elements that are present in both sets.
- [tex]\( A = \{1, 2, 3, 4, 6, 8, 12, 24\} \)[/tex]
- [tex]\( B = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24\} \)[/tex]
The common elements are [tex]\( \{2, 4, 6, 8, 12, 24\} \)[/tex]:
[tex]\[ A \cap B = \{2, 4, 6, 8, 12, 24\} \][/tex]
### d) What is [tex]\( A \cap C \)[/tex]?
The intersection of sets [tex]\( A \)[/tex] and [tex]\( C \)[/tex], denoted [tex]\( A \cap C \)[/tex], includes all elements that are present in both sets.
- [tex]\( A = \{1, 2, 3, 4, 6, 8, 12, 24\} \)[/tex]
- [tex]\( C = \{2, 4, 6, 8, 10, 12\} \)[/tex]
The common elements are [tex]\( \{2, 4, 6, 8, 12\} \)[/tex]:
[tex]\[ A \cap C = \{2, 4, 6, 8, 12\} \][/tex]
### e) What is [tex]\( A \cup B \)[/tex]?
The union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted [tex]\( A \cup B \)[/tex], includes all elements that are present in either set or both sets.
- [tex]\( A = \{1, 2, 3, 4, 6, 8, 12, 24\} \)[/tex]
- [tex]\( B = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24\} \)[/tex]
The combined set is:
[tex]\[ A \cup B = \{1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24\} \][/tex]
### f) What is [tex]\( A \cup C \)[/tex]?
The union of sets [tex]\( A \)[/tex] and [tex]\( C \)[/tex], denoted [tex]\( A \cup C \)[/tex], includes all elements that are present in either set or both sets.
- [tex]\( A = \{1, 2, 3, 4, 6, 8, 12, 24\} \)[/tex]
- [tex]\( C = \{2, 4, 6, 8, 10, 12\} \)[/tex]
The combined set is:
[tex]\[ A \cup C = \{1, 2, 3, 4, 6, 8, 10, 12, 24\} \][/tex]
### g) What is [tex]\( A^c \)[/tex]?
The complement of set [tex]\( A \)[/tex] in the universal set [tex]\( U \)[/tex], denoted [tex]\( A^c \)[/tex], includes all elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex].
- [tex]\( A = \{1, 2, 3, 4, 6, 8, 12, 24\} \)[/tex]
- [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24\} \)[/tex]
The elements in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex] are:
[tex]\[ A^c = \{5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23\} \][/tex]
To summarize:
a) [tex]\( C \subseteq B \)[/tex] : True
c) [tex]\( A \cap B = \{2, 4, 6, 8, 12, 24\} \)[/tex]
d) [tex]\( A \cap C = \{2, 4, 6, 8, 12\} \)[/tex]
e) [tex]\( A \cup B = \{1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24\} \)[/tex]
f) [tex]\( A \cup C = \{1, 2, 3, 4, 6, 8, 10, 12, 24\} \)[/tex]
g) [tex]\( A^c = \{5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23\} \)[/tex]
