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Two sets have m and n elements. The number of subsets of the first set is 112 more than the second set. The values of m and n are-

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Last updated date: 13th Jul 2024
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Answer
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Hint: We can solve this using basic sets concepts. First we calculate the number of subsets for each set and then we will derive the equation from the statement given in the question using the number of subsets of each set. After that we will simplify it until we arrive at the solution.

Complete step by step answer:
Given we have two sets m and n.
Generally a set containing n elements will have \[{{2}^{n}}\] subsets.
Using this we can find the number of subsets for our question.
We will get
Number of subsets for the \[{{1}^{st}}\] set having m elements is \[{{2}^{m}}\].
Now the number of subsets for the \[{{2}^{nd}}\] set having n elements is \[{{2}^{n}}\].
 Given that the number of subsets in the first set are 112 more than the second set.
From this we can write that
\[{{2}^{m}}-{{2}^{n}}=112\]
Now we have to simplify it to get the result.
Now we will multiply and divide the \[{{2}^{n}}\] on the LHS side. We will get
\[\Rightarrow \dfrac{{{2}^{m}}\times {{2}^{n}}}{{{2}^{n}}}-\dfrac{{{2}^{n}}\times {{2}^{n}}}{{{2}^{n}}}=112\]
We know that\[\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\].
Using this we can simplify the above expression.
By simplifying we will get
\[\Rightarrow \left( {{2}^{m-n}}\times {{2}^{n}} \right)-{{2}^{n}}=112\]
Now we take common terms out. So we will take \[{{2}^{n}}\] as common on the LHS side.
We will get
 \[\Rightarrow {{2}^{n}}\left( {{2}^{m-n}}-1 \right)=112\]
Now we have to write 112 as powers of 2 so that we can compare the terms and find the values.
We can write 112 as
\[112={{2}^{7}}-{{2}^{4}}\]
Using this the equation will become
\[\Rightarrow {{2}^{n}}\left( {{2}^{m-n}}-1 \right)={{2}^{7}}-{{2}^{4}}\]
Now we will take \[{{2}^{4}}\] as common on the RHS side.
We will get
\[\Rightarrow {{2}^{n}}\left( {{2}^{m-n}}-1 \right)={{2}^{4}}\left( {{2}^{3}}-1 \right)\]
Now we have to compare the terms as both RHS and LHS are looking the same.
By comparing we will get
\[{{2}^{n}}={{2}^{4}}\]
\[{{2}^{m-n}}={{2}^{3}}\]
Using this we can write
\[\Rightarrow n=4\] and
\[\Rightarrow m-n=3\]
Substituting n value we will get
\[\Rightarrow m-4=3\]
Now we add 4 on both sides and simplify it we will get
\[\Rightarrow m=3+4\]
So the value of m is
\[\Rightarrow m=7\]

The values of m and n are 7 and 4 respectively.

Note: we have to be careful while rewriting and simplifying the equation because if there is any change then we will get a different answer. Also we have to be aware of power formulas to solve these types of questions.