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16. A and B are subsets of a universal set [tex]U[/tex].

If [tex]U = \{x \mid x[/tex] is a natural number [tex]x \leqslant 10\}[/tex], [tex]A = \{y \mid y[/tex] is an odd number less than 10\}[/tex], and [tex]B = \{z \mid z[/tex] is a prime number less than 10\}[/tex], answer the following questions:

i. List the elements of [tex]A \cap B[/tex].

ii. What is the cardinal number of [tex]A - B[/tex]?

iii. Find [tex]\overline{A \cup B}[/tex] and illustrate it in a Venn diagram by shading.

iv. What is the relation between the sets [tex]A - B[/tex] and [tex]A[/tex]? Give a reason.

Sagot :

GinnyAnswer
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].
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