Answered

Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.

If [tex]U=\{x \mid 1 \leq x \leq 10, x \in \mathbb{N}\}[/tex], [tex]A=\{x \mid x[/tex] is a prime number [tex]\}[/tex], [tex]B=\{x \mid x[/tex] is an odd number [tex]\}[/tex], [tex]C=\{x \mid x[/tex] is divisible by 4 [tex]\}[/tex], find:

a) [tex](A \cup B)^{\prime}[/tex]

b) [tex](A \cap B)[/tex]

c) [tex](A \cap B) - (A \cup B)[/tex]

Sagot :

GinnyAnswer
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.
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.