Let's address each part of the problem step-by-step:
a) Description of the Universal Set [tex]\( U \)[/tex]
Given that the Universal Set [tex]\( U \)[/tex] includes 25 and needs to encompass all elements from the subsets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we can deduce that [tex]\( U \)[/tex] must include all whole numbers from 0 up to and including 25.
Therefore, in description method, the universal set [tex]\( U \)[/tex] is:
[tex]\[ U = \{ x \,|\, 0 \leq x \leq 25 \} \][/tex]
Subsets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] in Roster Method
- Subset [tex]\( A \)[/tex]
The subset [tex]\( A \)[/tex] consists of whole numbers less than 10. In roster method, it is written as:
[tex]\[ A = \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \][/tex]
- Subset [tex]\( B \)[/tex]
The subset [tex]\( B \)[/tex] consists of the factors of 24. In roster method, it is:
[tex]\[ B = \{ 1, 2, 3, 4, 6, 8, 12, 24 \} \][/tex]
Disjoint or Overlapping Subsets
To determine whether subsets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are disjoint or overlapping, we need to see if they have any elements in common. By examining the elements:
- Subset [tex]\( A \)[/tex]: [tex]\( \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \)[/tex]
- Subset [tex]\( B \)[/tex]: [tex]\( \{ 1, 2, 3, 4, 6, 8, 12, 24 \} \)[/tex]
We find the common elements: [tex]\( \{1, 2, 3, 4, 6, 8\} \)[/tex].
Since there are common elements between subsets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], they are overlapping. Therefore, the subsets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are not disjoint, because they share some elements.
Elements of Subset [tex]\( S \)[/tex]
The subset [tex]\( S \)[/tex] is defined as the set of square numbers that are elements of [tex]\( U \)[/tex]. Considering [tex]\( U = \{ x \,|\, 0 \leq x \leq 25 \} \)[/tex], we list all the square numbers between 0 and 25:
- [tex]\( 0^2 = 0 \)[/tex]
- [tex]\( 1^2 = 1 \)[/tex]
- [tex]\( 2^2 = 4 \)[/tex]
- [tex]\( 3^2 = 9 \)[/tex]
- [tex]\( 4^2 = 16 \)[/tex]
- [tex]\( 5^2 = 25 \)[/tex]
Hence, the subset [tex]\( S \)[/tex], containing these square numbers, is:
[tex]\[ S = \{ 0, 1, 4, 9, 16, 25 \} \][/tex]
By following this systematic approach, we have described and analyzed the given sets and subsets properly, addressing all parts of the problem comprehensively.
