Question

Two finite sets have m and k elements. If the total number of subsets of first set is 56 more than total number of subsets of second set, then find the value of m and k.

Answer 1

Hint: First, before proceeding for this, we must know that the total number of elements of a set containing n elements is represented by ${{2}^{n}}$. Then, using this property of sets, we get the total number of elements of the first set and second set. Then, we are given the condition in the question that a total number of sets of the first set is 56 more than a total number of sets of the second set and by using it, we get the final answer.

Complete step by step answer:
In this question, we are supposed to find the value of m and k when two finite sets have m and k elements and the total number of subsets of the first set is 56 more than the total number of subsets of the second set
So, before proceeding for this, we must know that the total number of elements of a set containing n elements is represented by:
${{2}^{n}}$
So, by using this property of sets, we get the total number of elements of the first set with m elements as:
${{2}^{m}}$
Similarly, by using this property of sets, we get the total number of elements of the second set with k elements as:
${{2}^{k}}$
Now, we are given the condition in the question that a total number of sets of the first set is 56 more than a total number of sets of the second set.
So, by using this condition, we get:
${{2}^{m}}-{{2}^{k}}=56$
Now, by taking ${{2}^{k}}$common from the left hand side and writing 56 as difference of 64 and 8 as:
$\begin{align}
  & {{2}^{k}}\left( {{2}^{m-k}}-1 \right)=64-8 \\
 & \Rightarrow {{2}^{k}}\left( {{2}^{m-k}}-1 \right)={{2}^{6}}-{{2}^{3}} \\
 & \Rightarrow {{2}^{k}}\left( {{2}^{m-k}}-1 \right)={{2}^{3}}\left( {{2}^{3}}-1 \right) \\
\end{align}$
So, by comparing the left hand side to right hand side, we get:
$\begin{align}
  & k=3 \\
 & m-k=3 \\
\end{align}$
Now, by substituting the value of k as 3 in the second result we found then:
$\begin{align}
  & m-3=3 \\
 & \Rightarrow m=3+3 \\
 & \Rightarrow m=6 \\
\end{align}$
Hence, we get the value of m and k as 6 and 3 respectively.

Note:
 Now, to solve these types of questions we need to know some of the basics of the exponents as we required the number 56 as the difference of the perfect power of 2 so that we can get the relation equivalent to the given conditions. So, we must be careful and aware of the power of 2 concepts.