Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
Let's address each part of the question one by one, using the information given:
### i. List the elements of [tex]\( A \cap B \)[/tex].
To start, let's define the sets:
- [tex]\( U \)[/tex] is the universal set [tex]\( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex]
- [tex]\( A \)[/tex] is the set of odd numbers less than 10, which is [tex]\( \{1, 3, 5, 7, 9\} \)[/tex]
- [tex]\( B \)[/tex] is the set of prime numbers less than 10, which includes [tex]\( \{2, 3, 5, 7\} \)[/tex]
To find [tex]\( A \cap B \)[/tex], we need the elements that are both in [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A \cap B = \{3, 5, 7\} \][/tex]
### ii. What is the cardinal number of [tex]\( A - B \)[/tex]?
[tex]\( A - B \)[/tex] represents the elements that are in [tex]\( A \)[/tex] but not in [tex]\( B \)[/tex]. So we need to find the elements in [tex]\( A \)[/tex] that are not in [tex]\( B \)[/tex]:
[tex]\[ A - B = \{1, 9\} \][/tex]
The cardinality is the number of elements in [tex]\( A - B \)[/tex]:
[tex]\[ \text{Cardinality of } A - B = 2 \][/tex]
### iii. Find [tex]\( \overline{A \cup B} \)[/tex] and illustrate it in a Venn diagram by shading.
First, we need to find [tex]\( A \cup B \)[/tex], which is the set of elements that are either in [tex]\( A \)[/tex] or in [tex]\( B \)[/tex] or in both:
[tex]\[ A \cup B = \{1, 2, 3, 5, 7, 9\} \][/tex]
The complement of [tex]\( A \cup B \)[/tex], denoted by [tex]\( \overline{A \cup B} \)[/tex], consists of all elements in the universal set [tex]\( U \)[/tex] that are not in [tex]\( A \cup B \)[/tex]:
[tex]\[ \overline{A \cup B} = \{4, 6, 8, 10\} \][/tex]
### iv. What is the relation between the sets [tex]\( A - B \)[/tex] and [tex]\( A \)[/tex]? Give reason.
By definition, [tex]\( A - B \)[/tex] contains elements that are in [tex]\( A \)[/tex] but not in [tex]\( B \)[/tex]. This means every element in [tex]\( A - B \)[/tex] is also in [tex]\( A \)[/tex]:
[tex]\[ A - B \subseteq A \][/tex]
Therefore, [tex]\( A - B \)[/tex] is a subset of [tex]\( A \)[/tex], confirming that:
[tex]\[ \{1, 9\} \subseteq \{1, 3, 5, 7, 9\} \][/tex]
This subset relation is inherent by the construction of set difference.
### Summary:
The detailed step-by-step solution yields:
i. [tex]\( A \cap B = \{3, 5, 7\} \)[/tex]
ii. The cardinal number of [tex]\( A - B \)[/tex] is 2.
iii. [tex]\( \overline{A \cup B} = \{4, 6, 8, 10\} \)[/tex]
iv. [tex]\( A - B \)[/tex] is a subset of [tex]\( A \)[/tex] because all elements of [tex]\( A - B \)[/tex] are taken from [tex]\( A \)[/tex].
### i. List the elements of [tex]\( A \cap B \)[/tex].
To start, let's define the sets:
- [tex]\( U \)[/tex] is the universal set [tex]\( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex]
- [tex]\( A \)[/tex] is the set of odd numbers less than 10, which is [tex]\( \{1, 3, 5, 7, 9\} \)[/tex]
- [tex]\( B \)[/tex] is the set of prime numbers less than 10, which includes [tex]\( \{2, 3, 5, 7\} \)[/tex]
To find [tex]\( A \cap B \)[/tex], we need the elements that are both in [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A \cap B = \{3, 5, 7\} \][/tex]
### ii. What is the cardinal number of [tex]\( A - B \)[/tex]?
[tex]\( A - B \)[/tex] represents the elements that are in [tex]\( A \)[/tex] but not in [tex]\( B \)[/tex]. So we need to find the elements in [tex]\( A \)[/tex] that are not in [tex]\( B \)[/tex]:
[tex]\[ A - B = \{1, 9\} \][/tex]
The cardinality is the number of elements in [tex]\( A - B \)[/tex]:
[tex]\[ \text{Cardinality of } A - B = 2 \][/tex]
### iii. Find [tex]\( \overline{A \cup B} \)[/tex] and illustrate it in a Venn diagram by shading.
First, we need to find [tex]\( A \cup B \)[/tex], which is the set of elements that are either in [tex]\( A \)[/tex] or in [tex]\( B \)[/tex] or in both:
[tex]\[ A \cup B = \{1, 2, 3, 5, 7, 9\} \][/tex]
The complement of [tex]\( A \cup B \)[/tex], denoted by [tex]\( \overline{A \cup B} \)[/tex], consists of all elements in the universal set [tex]\( U \)[/tex] that are not in [tex]\( A \cup B \)[/tex]:
[tex]\[ \overline{A \cup B} = \{4, 6, 8, 10\} \][/tex]
### iv. What is the relation between the sets [tex]\( A - B \)[/tex] and [tex]\( A \)[/tex]? Give reason.
By definition, [tex]\( A - B \)[/tex] contains elements that are in [tex]\( A \)[/tex] but not in [tex]\( B \)[/tex]. This means every element in [tex]\( A - B \)[/tex] is also in [tex]\( A \)[/tex]:
[tex]\[ A - B \subseteq A \][/tex]
Therefore, [tex]\( A - B \)[/tex] is a subset of [tex]\( A \)[/tex], confirming that:
[tex]\[ \{1, 9\} \subseteq \{1, 3, 5, 7, 9\} \][/tex]
This subset relation is inherent by the construction of set difference.
### Summary:
The detailed step-by-step solution yields:
i. [tex]\( A \cap B = \{3, 5, 7\} \)[/tex]
ii. The cardinal number of [tex]\( A - B \)[/tex] is 2.
iii. [tex]\( \overline{A \cup B} = \{4, 6, 8, 10\} \)[/tex]
iv. [tex]\( A - B \)[/tex] is a subset of [tex]\( A \)[/tex] because all elements of [tex]\( A - B \)[/tex] are taken from [tex]\( A \)[/tex].
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
