Answer :
Certainly! Let's address each part of the question in a detailed, step-by-step manner:
### Given Sets:
1. Universal Set [tex]\( U \)[/tex]:
- [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex]
- This set includes all natural numbers from 1 to 10.
2. Set [tex]\( A \)[/tex]:
- [tex]\( A = \{x \mid x \text{ is a prime number less than 10}\} \)[/tex]
- Prime numbers less than 10 are: [tex]\( A = \{2, 3, 5, 7\} \)[/tex].
3. Set [tex]\( B \)[/tex]:
- [tex]\( B = \{x \mid x \text{ is an odd number less than 10}\} \)[/tex]
- Odd numbers less than 10 are: [tex]\( B = \{1, 3, 5, 7, 9\} \)[/tex].
4. Set [tex]\( C \)[/tex]:
- [tex]\( C = \{x \mid x \text{ is divisible by 4 less than 10}\} \)[/tex]
- Numbers divisible by 4 less than 10 are: [tex]\( C = \{4, 8\} \)[/tex].
### Part a) Calculate [tex]\((A \cup B)^{\prime}\)[/tex]:
1. Union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] ( [tex]\( A \cup B \)[/tex] ):
- [tex]\( A \cup B = \{2, 3, 5, 7\} \cup \{1, 3, 5, 7, 9\} \)[/tex]
- [tex]\( A \cup B = \{1, 2, 3, 5, 7, 9\} \)[/tex]
2. Complement of [tex]\( A \cup B \)[/tex] in [tex]\( U \)[/tex] ( [tex]\( (A \cup B)^{\prime} \)[/tex] ):
- [tex]\( (A \cup B)^{\prime} = U - (A \cup B) \)[/tex]
- [tex]\( (A \cup B)^{\prime} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} - \{1, 2, 3, 5, 7, 9\} \)[/tex]
- [tex]\( (A \cup B)^{\prime} = \{4, 6, 8, 10\} \)[/tex]
Therefore, [tex]\((A \cup B)^{\prime} = \{4, 6, 8, 10\}\)[/tex].
### Part b) Calculate [tex]\( A \cap B \)[/tex]:
1. Intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] ( [tex]\( A \cap B \)[/tex] ):
- [tex]\( A \cap B = \{2, 3, 5, 7\} \cap \{1, 3, 5, 7, 9\} \)[/tex]
- [tex]\( A \cap B = \{3, 5, 7\} \)[/tex]
Therefore, [tex]\( A \cap B = \{3, 5, 7\} \)[/tex].
### Part c) Calculate [tex]\((A \cap B) - (A \cup B)\)[/tex]:
1. Difference between [tex]\( A \cap B \)[/tex] and [tex]\( A \cup B \)[/tex] [tex]\( ( (A \cap B) - (A \cup B) ) \)[/tex]:
- [tex]\( (A \cap B) - (A \cup B) = \{3, 5, 7\} - \{1, 2, 3, 5, 7, 9\} \)[/tex]
- The elements in [tex]\(\{3, 5, 7\}\)[/tex] intersect with [tex]\(\{1, 2, 3, 5, 7, 9\}\)[/tex], leading to an empty set.
- [tex]\( (A \cap B) - (A \cup B) = \emptyset \)[/tex]
Therefore, [tex]\((A \cap B) - (A \cup B) = \emptyset \)[/tex].
### Final Results:
- [tex]\((A \cup B)^{\prime} = \{4, 6, 8, 10\}\)[/tex]
- [tex]\( A \cap B = \{3, 5, 7\} \)[/tex]
- [tex]\( (A \cap B) - (A \cup B) = \emptyset \)[/tex]
These are the derived solutions for each part of the question.
### Given Sets:
1. Universal Set [tex]\( U \)[/tex]:
- [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex]
- This set includes all natural numbers from 1 to 10.
2. Set [tex]\( A \)[/tex]:
- [tex]\( A = \{x \mid x \text{ is a prime number less than 10}\} \)[/tex]
- Prime numbers less than 10 are: [tex]\( A = \{2, 3, 5, 7\} \)[/tex].
3. Set [tex]\( B \)[/tex]:
- [tex]\( B = \{x \mid x \text{ is an odd number less than 10}\} \)[/tex]
- Odd numbers less than 10 are: [tex]\( B = \{1, 3, 5, 7, 9\} \)[/tex].
4. Set [tex]\( C \)[/tex]:
- [tex]\( C = \{x \mid x \text{ is divisible by 4 less than 10}\} \)[/tex]
- Numbers divisible by 4 less than 10 are: [tex]\( C = \{4, 8\} \)[/tex].
### Part a) Calculate [tex]\((A \cup B)^{\prime}\)[/tex]:
1. Union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] ( [tex]\( A \cup B \)[/tex] ):
- [tex]\( A \cup B = \{2, 3, 5, 7\} \cup \{1, 3, 5, 7, 9\} \)[/tex]
- [tex]\( A \cup B = \{1, 2, 3, 5, 7, 9\} \)[/tex]
2. Complement of [tex]\( A \cup B \)[/tex] in [tex]\( U \)[/tex] ( [tex]\( (A \cup B)^{\prime} \)[/tex] ):
- [tex]\( (A \cup B)^{\prime} = U - (A \cup B) \)[/tex]
- [tex]\( (A \cup B)^{\prime} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} - \{1, 2, 3, 5, 7, 9\} \)[/tex]
- [tex]\( (A \cup B)^{\prime} = \{4, 6, 8, 10\} \)[/tex]
Therefore, [tex]\((A \cup B)^{\prime} = \{4, 6, 8, 10\}\)[/tex].
### Part b) Calculate [tex]\( A \cap B \)[/tex]:
1. Intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] ( [tex]\( A \cap B \)[/tex] ):
- [tex]\( A \cap B = \{2, 3, 5, 7\} \cap \{1, 3, 5, 7, 9\} \)[/tex]
- [tex]\( A \cap B = \{3, 5, 7\} \)[/tex]
Therefore, [tex]\( A \cap B = \{3, 5, 7\} \)[/tex].
### Part c) Calculate [tex]\((A \cap B) - (A \cup B)\)[/tex]:
1. Difference between [tex]\( A \cap B \)[/tex] and [tex]\( A \cup B \)[/tex] [tex]\( ( (A \cap B) - (A \cup B) ) \)[/tex]:
- [tex]\( (A \cap B) - (A \cup B) = \{3, 5, 7\} - \{1, 2, 3, 5, 7, 9\} \)[/tex]
- The elements in [tex]\(\{3, 5, 7\}\)[/tex] intersect with [tex]\(\{1, 2, 3, 5, 7, 9\}\)[/tex], leading to an empty set.
- [tex]\( (A \cap B) - (A \cup B) = \emptyset \)[/tex]
Therefore, [tex]\((A \cap B) - (A \cup B) = \emptyset \)[/tex].
### Final Results:
- [tex]\((A \cup B)^{\prime} = \{4, 6, 8, 10\}\)[/tex]
- [tex]\( A \cap B = \{3, 5, 7\} \)[/tex]
- [tex]\( (A \cap B) - (A \cup B) = \emptyset \)[/tex]
These are the derived solutions for each part of the question.
