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Sagot :
Let's break down and solve each part of the given problem step by step.
### Step-by-Step Solution
#### a) Given:
- Universal set [tex]\( U = \{ x \mid x \text{ is a natural number}, x \leq 10 \} \)[/tex].
- Set [tex]\( A = \{ y \mid y \text{ is an odd number less than 10} \} \)[/tex].
- Set [tex]\( B = \{ z \mid z \text{ is a prime number less than 10} \} \)[/tex].
We first determine the actual sets based on these descriptions.
- [tex]\( U \)[/tex] contains natural numbers less than or equal to 10: [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex].
- [tex]\( A \)[/tex] contains odd numbers less than 10: [tex]\( A = \{1, 3, 5, 7, 9\} \)[/tex].
- [tex]\( B \)[/tex] contains prime numbers less than 10: [tex]\( B = \{2, 3, 5, 7\} \)[/tex].
Now, we proceed with each question.
#### i. List the elements of [tex]\( A \cap B \)[/tex]:
The intersection [tex]\( A \cap B \)[/tex] includes elements that are common to both sets [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]?
The set difference [tex]\( A - B \)[/tex] includes elements that are in [tex]\( A \)[/tex] but not in [tex]\( B \)[/tex]:
[tex]\[ A = \{1, 3, 5, 7, 9\} \][/tex]
[tex]\[ B = \{2, 3, 5, 7\} \][/tex]
[tex]\[ A - B = \{1, 9\} \][/tex]
The cardinal number (or the size) of the set [tex]\( A - B \)[/tex] is the number of elements in it:
[tex]\[ \text{Cardinal number of } A - B = 2 \][/tex]
#### iii. Find [tex]\( \triangle \cup B \)[/tex] and illustrate it in a Venn diagram by shading.
The symmetric difference [tex]\( A \triangle B \)[/tex] of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is defined as the set of elements that are in [tex]\( A \)[/tex] or [tex]\( B \)[/tex] but not in their intersection.
[tex]\[ A \triangle B = (A - B) \cup (B - A) \][/tex]
[tex]\[ A \triangle B = \{1, 9\} \cup \{2\} = \{1, 2, 9\} \][/tex]
The union [tex]\( \triangle \cup B \)[/tex] includes all elements from the symmetric difference [tex]\( A \triangle B \)[/tex] and elements of [tex]\( B \)[/tex]:
[tex]\[ A \triangle B = \{1, 2, 9\} \][/tex]
Thus, the result remains the symmetric difference as the elements of [tex]\( B \)[/tex] that are not in [tex]\( A \triangle B\)[/tex] are already included.
In a Venn diagram, the symmetric difference (shaded area) would include the elements that are either in [tex]\( A \)[/tex] or [tex]\( B \)[/tex] but not in both.
#### iv. What is the relation between the sets [tex]\( A - B \)[/tex] and [tex]\( A \)[/tex]? Give reason.
To determine the relation, observe the definition of set difference [tex]\( A - B \)[/tex]:
[tex]\[ A - B = \{1, 9\} \][/tex]
Clearly, all elements in [tex]\( A - B \)[/tex] are also elements of [tex]\( A \)[/tex], so:
[tex]\[ A - B \subseteq A \][/tex]
Hence, the set [tex]\( A - B \)[/tex] is a subset of [tex]\( A \)[/tex] and contains elements of [tex]\( A \)[/tex] that are not in [tex]\( B \)[/tex]. Therefore, the relation is:
[tex]\[ A - B \text{ is a subset of } A \text{ and contains elements of } A \text{ that are not in } B. \][/tex]
### Summary:
i. [tex]\( A \cap B = \{3, 5, 7\} \)[/tex]
ii. The cardinal number of [tex]\( A - B \)[/tex] is 2.
iii. [tex]\( A \triangle B = \{1, 2, 9\} \)[/tex]
iv. [tex]\( A - B \text{ is a subset of } A \text{ and contains elements of } A \text{ that are not in } B. \)[/tex]
These are the detailed step-by-step solutions for the given questions.
### Step-by-Step Solution
#### a) Given:
- Universal set [tex]\( U = \{ x \mid x \text{ is a natural number}, x \leq 10 \} \)[/tex].
- Set [tex]\( A = \{ y \mid y \text{ is an odd number less than 10} \} \)[/tex].
- Set [tex]\( B = \{ z \mid z \text{ is a prime number less than 10} \} \)[/tex].
We first determine the actual sets based on these descriptions.
- [tex]\( U \)[/tex] contains natural numbers less than or equal to 10: [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex].
- [tex]\( A \)[/tex] contains odd numbers less than 10: [tex]\( A = \{1, 3, 5, 7, 9\} \)[/tex].
- [tex]\( B \)[/tex] contains prime numbers less than 10: [tex]\( B = \{2, 3, 5, 7\} \)[/tex].
Now, we proceed with each question.
#### i. List the elements of [tex]\( A \cap B \)[/tex]:
The intersection [tex]\( A \cap B \)[/tex] includes elements that are common to both sets [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]?
The set difference [tex]\( A - B \)[/tex] includes elements that are in [tex]\( A \)[/tex] but not in [tex]\( B \)[/tex]:
[tex]\[ A = \{1, 3, 5, 7, 9\} \][/tex]
[tex]\[ B = \{2, 3, 5, 7\} \][/tex]
[tex]\[ A - B = \{1, 9\} \][/tex]
The cardinal number (or the size) of the set [tex]\( A - B \)[/tex] is the number of elements in it:
[tex]\[ \text{Cardinal number of } A - B = 2 \][/tex]
#### iii. Find [tex]\( \triangle \cup B \)[/tex] and illustrate it in a Venn diagram by shading.
The symmetric difference [tex]\( A \triangle B \)[/tex] of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is defined as the set of elements that are in [tex]\( A \)[/tex] or [tex]\( B \)[/tex] but not in their intersection.
[tex]\[ A \triangle B = (A - B) \cup (B - A) \][/tex]
[tex]\[ A \triangle B = \{1, 9\} \cup \{2\} = \{1, 2, 9\} \][/tex]
The union [tex]\( \triangle \cup B \)[/tex] includes all elements from the symmetric difference [tex]\( A \triangle B \)[/tex] and elements of [tex]\( B \)[/tex]:
[tex]\[ A \triangle B = \{1, 2, 9\} \][/tex]
Thus, the result remains the symmetric difference as the elements of [tex]\( B \)[/tex] that are not in [tex]\( A \triangle B\)[/tex] are already included.
In a Venn diagram, the symmetric difference (shaded area) would include the elements that are either in [tex]\( A \)[/tex] or [tex]\( B \)[/tex] but not in both.
#### iv. What is the relation between the sets [tex]\( A - B \)[/tex] and [tex]\( A \)[/tex]? Give reason.
To determine the relation, observe the definition of set difference [tex]\( A - B \)[/tex]:
[tex]\[ A - B = \{1, 9\} \][/tex]
Clearly, all elements in [tex]\( A - B \)[/tex] are also elements of [tex]\( A \)[/tex], so:
[tex]\[ A - B \subseteq A \][/tex]
Hence, the set [tex]\( A - B \)[/tex] is a subset of [tex]\( A \)[/tex] and contains elements of [tex]\( A \)[/tex] that are not in [tex]\( B \)[/tex]. Therefore, the relation is:
[tex]\[ A - B \text{ is a subset of } A \text{ and contains elements of } A \text{ that are not in } B. \][/tex]
### Summary:
i. [tex]\( A \cap B = \{3, 5, 7\} \)[/tex]
ii. The cardinal number of [tex]\( A - B \)[/tex] is 2.
iii. [tex]\( A \triangle B = \{1, 2, 9\} \)[/tex]
iv. [tex]\( A - B \text{ is a subset of } A \text{ and contains elements of } A \text{ that are not in } B. \)[/tex]
These are the detailed step-by-step solutions for the given questions.
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