Sure, let's determine whether the number of proper subsets for the given sets \( x = \{0, 5\} \), \( y = \{11, 5\} \), and \( z = \{1, 4\} \) are equal.
Firstly, it is essential to understand the concept of proper subsets. A proper subset of a set is any subset of that set which is not equal to the set itself.
For a set with \( n \) elements, the total number of subsets is \( 2^n \). The number of proper subsets, however, is \( 2^n - 1 \), because you exclude the set itself from the total number of subsets.
Now, let's determine this step-by-step:
### Step 1: Determine the number of elements in each set
- Set \( x = \{0, 5\} \) has 2 elements.
- Set \( y = \{11, 5\} \) has 2 elements.
- Set \( z = \{1, 4\} \) has 2 elements.
### Step 2: Calculate the number of proper subsets for each set
Using the formula \( 2^n - 1 \):
- For set \( x \), with 2 elements, the number of proper subsets is \( 2^2 - 1 = 4 - 1 = 3 \).
- For set \( y \), with 2 elements, the number of proper subsets is \( 2^2 - 1 = 4 - 1 = 3 \).
- For set \( z \), with 2 elements, the number of proper subsets is \( 2^2 - 1 = 4 - 1 = 3 \).
### Conclusion
The number of proper subsets for the sets \( x \), \( y \), and \( z \) are all equal and each has 3 proper subsets. This shows consistently that regardless of the elements themselves, as long as the sets have the same number of elements, the count of proper subsets will be the same.
Thus, yes, the number of proper subsets of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] are equal, each with 3 proper subsets. This validates our understanding of subsets in set theory.
