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Given the sets:

[tex]\[ A=\{x \mid x \text{ is a whole number less than 10}\} \][/tex]
[tex]\[ B=\{\text{factors of 24}\} \][/tex]

which are subsets of a universal set [tex]\( U \)[/tex].

a) If [tex]\( 25 \notin U \)[/tex], write the universal set [tex]\( U \)[/tex] using the description method.

b) Write the subsets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] using the roster method.

c) Are the subsets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] disjoint or overlapping? Give a reason.

Sagot :

GinnyAnswer
Certainly! Let's solve the problem step-by-step:

### Given:
- A = \{x: x is a whole number less than 10\}
- B = \{factors of 24\}
- It is also given that 25 is not an element of the universal set [tex]\(U\)[/tex].

### Part (a): Universal Set [tex]\(U\)[/tex] in Description Method
The universal set [tex]\(U\)[/tex] consists of all whole numbers less than 25. So, the universal set in the description method is:
[tex]\[ U = \{x: x \text{ is a whole number less than 25}\} \][/tex]

### Part (b): Subsets [tex]\(A\)[/tex] and [tex]\(B\)[/tex] in Roster Method
Now, let's list the elements of subsets [tex]\(A\)[/tex] and [tex]\(B\)[/tex].

#### Subset [tex]\(A\)[/tex] in Roster Method
Since [tex]\(A\)[/tex] is defined as the set of whole numbers less than 10:
[tex]\[ A = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \][/tex]

#### Subset [tex]\(B\)[/tex] in Roster Method
[tex]\(B\)[/tex] is defined as the set of factors of 24. The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.
[tex]\[ B = \{1, 2, 3, 4, 6, 8, 12, 24\} \][/tex]

### Part (c): Are Subsets [tex]\(A\)[/tex] and [tex]\(B\)[/tex] Disjoint or Overlapping?
To determine whether [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are disjoint or overlapping, we look at their common elements.

- Common Elements:
[tex]\[ A \cap B = \{1, 2, 3, 4, 6, 8\} \][/tex]

The common elements between [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are: 1, 2, 3, 4, 6, and 8. Since there are elements that are present in both subsets, we can conclude that:

- Observation:
Since [tex]\(A\)[/tex] and [tex]\(B\)[/tex] have common elements, they are overlapping.

### Conclusion
- [tex]\(U = \{x: x \text{ is a whole number less than 25}\}\)[/tex]
- [tex]\(A = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\)[/tex]
- [tex]\(B = \{1, 2, 3, 4, 6, 8, 12, 24\}\)[/tex]
- Subsets [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are overlapping because they share common elements: \{1, 2, 3, 4, 6, 8\}

That concludes our detailed step-by-step solution for the given question.
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