Two finite sets have n and m elements. if the total number of subsets of the first set is 56 more than the second set then find the values of m and n
Again... the first hit of your exact question gave a well layed out explanation....
It went like this.
Let M and N be two sets having m and n numbers of elements. The number of subsets of M = 2m and the number of subsets of N =2n.
The question stated that they subsets differ by 56. Written mathematically that is, 2m - 2n = 56
...
Since the number of subsets of k elements is 2^k ,
I think you mean
2^m - 2^n = 56
by inspection, the powers of 2 are
1, 2, 4, 8, 16, 32, 64, 128, 256, ....
so we want a difference of two of these of 56
No need to go higher than 64 ...
64 - 16 = 48, not what we want
64 - 8 = 56 , THAT'S THE ONE
3^6 - 2^3 = 56
so m = 6 and n = 3
To find the values of m and n, we can use combinatorics principles.
Let's denote the number of elements in the first set as n and the number of elements in the second set as m.
The number of subsets of a set with n elements is 2^n, and the number of subsets of a set with m elements is 2^m.
According to the given information, the total number of subsets of the first set is 56 more than the total number of subsets of the second set. Mathematically, we can represent this as:
2^n = 2^m + 56
To solve this equation, we can use logarithms. Taking the logarithm base 2 of both sides gives:
n = m + log2(2^m + 56)
Since logarithms might not yield whole number answers, we need to find values of m that satisfy this equation and where m and n are both positive integers.
To find the values of m and n that satisfy the equation, we can try different values of m and see if it gives a corresponding positive integer value for n.
For example, let's start by assuming m as 1 and solving for n:
n = 1 + log2(2^1 + 56) = 1 + log2(58)
Here, log2(58) is approximately 5.858. So n is not an integer in this case.
Continuing this process, we need to try different values of m until we find a pair of m and n that satisfies the equation and where both m and n are positive integers.