Sets, Relations & Functions
Power Set
Grade 11
Question:
<p>Set A has m elements and Set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of \(m \times n\) is ___________.</p><p>(JEE Main 2020)</p>
Step-by-Step Solution
Key Concept: The number of subsets of a set with k elements is $2^k$. Set up an equation using this property and factor to find m and n.
<p><strong>Step 1:</strong> Set up the equation using the number of subsets.</p><p>It is given that $n(A) = m$ and $n(B) = n$.</p><p>The number of subsets of set A is $2^m$ and the number of subsets of set B is $2^n$.</p><p><strong>Step 2:</strong> Form the equation.</p><p>$$ 2^m = 2^n + 112 $$</p><p>$$ 2^m - 2^n = 112 $$</p><p><strong>Step 3:</strong> Factor the left side.</p><p>$$ 2^m - 2^n = 2^4 \times 7 $$</p><p>$$ 2^n(2^{m-n} - 1) = 2^4(2^3 - 1) $$</p><p><strong>Step 4:</strong> Compare the exponents and values.</p><p>On comparing: $n = 4$ and $m - n = 3$</p><p>Therefore: $m = 7$</p><p><strong>Step 5:</strong> Calculate the product.</p><p>$$ m \times n = 7 \times 4 = 28 $$</p>
Correct Answer: 28