Question

Two finite sets $A$ and $B$ have $m$ and $n$ elements respectively. If the total number of subsets of $A$ is $112$ more than the total number of subsets of $B$, then the value of m is:
A. $7$
B. $9$
C. $10$
D. $12$
E. $13$

Answer 1

In the given question, we are provided with two finite sets $A$ and $B$. The number of elements in both the sets is given to us in the form of variables $m$ and $n$. We are also given that the number of subsets of set $A$ is greater than the number of subsets of set $B$ by a specific number. So, we will first calculate the difference in the total number of subsets for the two sets and then find the value of $m$ and $n$.

Complete step by step answer:
So, the total number of elements in set $A$ is $m$. Total number of elements in set $B$ is $n$. Now, we know that the total number of subsets of a given set which consists of $p$ elements in total is ${2^p}$. So, the total number of subsets of set $A$ is ${2^m}$. Also, the total number of subsets of set $B$ is ${2^n}$.

Now, we are given that the total number of subsets of $A$ is $112$ more than the total number of subsets of $B$. So, $n$ must be less than m as the number of subsets of set $B$ is less than the number of subsets of set $A$.
So, we get, ${2^m} - {2^n} = 112$.
Now, taking ${2^n}$ common from left side of the equation, we get,
$\Rightarrow {2^n}\left( {\dfrac{{{2^m}}}{{{2^n}}} - 1} \right) = 112$
Now, we factorize $112$ in the right side of the equation. Also, using the exponent law $\dfrac{{{a^x}}}{{{a^y}}} = {a^{x - y}}$ in the equation, we get,
$\Rightarrow {2^n}\left( {{2^{\left( {m - n} \right)}} - 1} \right) = {2^4} \times 7$
Now, we know that any power of two can only be equal to a power of two as it consists only of $2$ as its factor. So, we equate ${2^n}$ with ${2^4}$. Hence, we will have to equate the factor $\left( {{2^{\left( {m - n} \right)}} - 1} \right)$ with $7$ so that the equation holds true. So, we get,
$\Rightarrow {2^n} = {2^4}$ and $\left( {{2^{\left( {m - n} \right)}} - 1} \right) = 7$

So, adding one to both sides of $\left( {{2^{\left( {m - n} \right)}} - 1} \right) = 7$, we get both the equation as,
$\Rightarrow {2^n} = {2^4}$ and ${2^{\left( {m - n} \right)}} = 7 + 1 = 8$
Now, we know that $8$ can be expressed as ${2^3}$.
$\Rightarrow {2^n} = {2^4}$ and ${2^{\left( {m - n} \right)}} = {2^3}$
Now, we compare the powers of two on both sides of the two equations. So, we get,
$\Rightarrow n = 4$ and $m - n = 3$
So, we get the value of n as $4$.
Now, we have to find the value of m. So, we put the value of n in the equation $m - n = 3$. So, we get,
$\Rightarrow m - 4 = 3$
Adding $4$ to both sides of the equation, we get,
$\Rightarrow m = 3 + 4$
$\therefore m = 7$
Hence, we get the value of $m$ as $7$.

Therefore, option A is the correct answer.

Note: We must know the formula for the total number of subsets of a given set in order to solve the given problem. One should know the simplification rules and algebraic rules like transposition to solve the equation that is formed while solving the problem. One must have a good knowledge of laws of exponents so as to get through with such types of problems.