Let's go through the problem step by step to find the solution.
### Part (a): Number of Subsets
First, we need to determine the set [tex]\( S \)[/tex].
The set [tex]\( S \)[/tex] is defined as:
[tex]\[ S = \{ x : x \text{ is a square number, } 1 \leq x < 5 \} \][/tex]
We need to find square numbers between 1 and 5:
- The square of 1 is [tex]\( 1^2 = 1 \)[/tex]
- The square of 2 is [tex]\( 2^2 = 4 \)[/tex]
Therefore, [tex]\( S = \{ 1, 4 \} \)[/tex].
Now, to find the number of subsets of a set, we use the formula for the number of subsets, which is [tex]\( 2^n \)[/tex], where [tex]\( n \)[/tex] is the number of elements in the set.
Here, [tex]\( n = 2 \)[/tex] because there are two elements (1 and 4) in the set [tex]\( S \)[/tex].
Thus, the number of subsets of [tex]\( S \)[/tex] is:
[tex]\[ 2^2 = 4 \][/tex]
### Part (b): List of All Subsets
Next, we list all possible subsets of [tex]\( S \)[/tex]. A subset can have any number of elements from the set, including zero elements.
The subsets of [tex]\( \{1, 4\} \)[/tex] are:
1. The empty set: [tex]\( \emptyset \)[/tex] or [tex]\( [] \)[/tex]
2. The single-element subsets:
- [tex]\( \{1\} \)[/tex]
- [tex]\( \{4\} \)[/tex]
3. The subset containing both elements: [tex]\( \{1, 4\} \)[/tex]
Listing them out, we get:
[tex]\[ \text{Subsets: } \{ \emptyset, \{1\}, \{4\}, \{1, 4\} \} \][/tex]
### Final Answer
(a) The number of subsets of [tex]\( S \)[/tex] is 4.
(b) The possible subsets of [tex]\( S \)[/tex] are:
[tex]\[ \{ [], [1], [4], [1, 4] \} \][/tex]
This concludes the detailed solution for the question.
