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
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
Updated On: Jun 7, 2024
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The Correct Option is A
Solution and Explanation
n(A) = m, n(B) = n
$P(A) = 2^m, P(B) = 2^n$
$\therefore 2^m-2^n=112=2\times 2 \times 2 \times 2 \times 7 $
$ \,2^n(2^{m-n}-1)=2^4 \times 7 =2^4(2^3-1)$
$\therefore n=4,m-n=3 \Rightarrow m-4 =3\Rightarrow m=7$.
$P(A) = 2^m, P(B) = 2^n$
$\therefore 2^m-2^n=112=2\times 2 \times 2 \times 2 \times 7 $
$ \,2^n(2^{m-n}-1)=2^4 \times 7 =2^4(2^3-1)$
$\therefore n=4,m-n=3 \Rightarrow m-4 =3\Rightarrow m=7$.
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Concepts Used:
Operations on Sets
Some important operations on sets include union, intersection, difference, and the complement of a set, a brief explanation of operations on sets is as follows:
1. Union of Sets:
- The union of sets lists the elements in set A and set B or the elements in both set A and set B.
- For example, {3,4} ∪ {1, 4} = {1, 3, 4}
- It is denoted as “A U B”
2. Intersection of Sets:
- Intersection of sets lists the common elements in set A and B.
- For example, {3,4} ∪ {1, 4} = {4}
- It is denoted as “A ∩ B”
3.Set Difference:
- Set difference is the list of elements in set A which is not present in set B
- For example, {3,4} - {1, 4} = {3}
- It is denoted as “A - B”
4.Set Complement:
- The set complement is the list of all elements present in the Universal set except the elements present in set A
- It is denoted as “U-A”