Let C = {p, l, u, s}. How many proper subsets does C have?
p,l,u,s,pl,pu,ps,lu,ls,us,plu,pls,pus,lus
check those and count them.
Most texts include the null set, or{}, as a subset of every set.
So for n elements in any set there would be 2^n - 1 proper subsets.
In your case there would be 15.
Notice we excluded the set itself, or {p,l,u,s}, from the list of 'proper' subsets
To find the number of proper subsets, we need to exclude the empty set and the set C itself. To do this, we can use the formula 2^n - 2, where n is the number of elements in the set.
In this case, the set C has 4 elements: p, l, u, and s. So, the number of proper subsets is 2^4 - 2 = 16 - 2 = 14.
Here's how you can get this answer:
1. Start with the set C = {p, l, u, s}.
2. Find the power set of C, which is the set of all subsets of C. To do this, you can consider each element of C and decide whether to include it or exclude it in each subset.
- Including an element means it is part of the subset.
- Excluding an element means it is not part of the subset.
3. Count the number of subsets you obtained from step 2.
- Exclude the empty set, which is the subset with no elements.
- Exclude the set C itself, which is the subset with all the elements of C.
4. The remaining count will give you the number of proper subsets.
In this case, from step 2, you will find 16 subsets: {}, {p}, {l}, {u}, {s}, {p, l}, {p, u}, {p, s}, {l, u}, {l, s}, {u, s}, {p, l, u}, {p, l, s}, {p, u, s}, {l, u, s}, {p, l, u, s}.
After excluding the empty set and the set C itself, you are left with 14 proper subsets.