Alright, let’s break this problem down step by step.
The given set [tex]\( S = \{ x \mid x \)[/tex] is a square number, [tex]\( 1 \leq x < 5 \} \)[/tex].
First, we need to identify the members of set [tex]\( S \)[/tex].
- The numbers between 1 and 4 that are squares are: 1 and 4.
- So, [tex]\( S = \{ 1, 4 \} \)[/tex].
### (a) How many subsets of [tex]\( S \)[/tex] are possible?
The number of subsets of a set with [tex]\( n \)[/tex] elements is given by [tex]\( 2^n \)[/tex].
Here, the set [tex]\( S \)[/tex] has 2 elements (1 and 4).
Therefore, the number of subsets is [tex]\( 2^2 = 4 \)[/tex].
### (b) Write all possible subsets of [tex]\( S \)[/tex].
To find all possible subsets, we list subsets with different numbers of elements:
1. Subset with 0 elements: [tex]\( \{ \} \)[/tex] (empty set)
2. Subsets with 1 element: [tex]\( \{ 1 \} \)[/tex], [tex]\( \{ 4 \} \)[/tex]
3. Subset with 2 elements: [tex]\( \{ 1, 4 \} \)[/tex]
So, all possible subsets of [tex]\( S \)[/tex] are:
[tex]\( \{ \}, \{1\}, \{4\}, \{1, 4\} \)[/tex].
### (c) Which one is the improper subset of [tex]\( S \)[/tex]? Give reason.
By definition, the improper subset of a set is the set itself.
Thus, the improper subset of [tex]\( S \)[/tex] is [tex]\( S \)[/tex] itself, which is [tex]\( \{ 1, 4 \} \)[/tex].
Reason: A subset is a set that contains some or all elements of the original set. The only subset that contains all elements of the original set [tex]\( S \)[/tex] (making it exactly equal to [tex]\( S \)[/tex]) is [tex]\( \{ 1, 4 \} \)[/tex]. Hence, [tex]\( \{ 1, 4 \} \)[/tex] is the improper subset.
