How many elements must a set have if the number of proper subsets of the set is 1/2 of the total number of subsets of the set?
Let's denote the number of elements in the set as "n".
The total number of subsets for a set with n elements is 2^n.
The number of proper subsets (excluding the empty set and the set itself) is 2^n - 2 (since there are 2^n subsets and 2 of them are not proper subsets).
According to the given condition, the number of proper subsets must be equal to half of the total number of subsets:
2^n - 2 = (1/2) * 2^n.
To solve this equation, we can multiply both sides by 2 to get rid of the fraction:
2 * (2^n - 2) = 2 * (1/2) * 2^n,
2^(n+1) - 4 = 2^n.
Next, we can subtract 2^n from both sides of the equation to isolate the power of 2 on one side:
2^(n+1) - 2^n - 4 = 0.
Now, we can factor out 2^n from the first two terms:
2^n * (2 - 1) - 4 = 0,
2^n - 4 = 0.
Adding 4 to both sides:
2^n = 4.
Taking the logarithm base 2 of both sides:
n = log2(4),
n = 2.
Therefore, a set must have 2 elements if the number of proper subsets is half the total number of subsets.
Let's solve this step by step:
1. Let's assume that the set has "n" elements.
2. The total number of subsets of a set with "n" elements is 2^n, including the empty set and the set itself.
3. The number of proper subsets (excluding the empty set and the set itself) would be 2^n - 2.
4. According to the problem, the number of proper subsets is 1/2 of the total number of subsets.
5. So, we have the equation 2^n - 2 = (1/2) * 2^n.
6. Simplifying the equation, we get 2^n - 2 = 2^(n-1).
7. Now, we can solve this equation:
2^n - 2 = 2^(n-1)
2^n - 2 = 2^n / 2
8. Multiplying both sides by 2, we get:
2^(n+1) - 4 = 2^n
9. Subtracting 2^n from both sides, we get:
2^(n+1) - 2^n - 4 = 0
10. We can simplify this equation further:
2^n(2-1) - 4 = 0
2^n - 4 = 0
11. Adding 4 to both sides:
2^n = 4
12. Taking the logarithm base 2 of both sides:
log2(2^n) = log2(4)
n = log2(4)
n = 2
13. Therefore, the set must have 2 elements.