Question

There are two sets \(A\) and \(B\) with \(|A|=m\) and \(|B|=n\). If \(|P(A)|-|P(B)|=112\), choose the wrong option.

• A. \(m+n=11\)
• B. \(2m-n=3\)
• C. \(2m-n=1\)
• D. \(3n-m=5\)

Hint:

Whenever a power set question appears, immediately replace \[ |P(A)|=2^{|A|}. \] Then convert the problem into an equation involving powers of 2.

Answer 1

The Correct Option is D

Step 1: Recall the power set size.
If a set has $k$ elements, its power set has $2^k$ subsets. So $|P(A)| = 2^m$ and $|P(B)| = 2^n$.

Step 2: Write the given equation.
\[ 2^m - 2^n = 112 \]
Step 3: Take out the common factor.
Pull $2^n$ out from both terms.
\[ 2^n\left(2^{m-n} - 1\right) = 112 \]
Step 4: Break 112 into a power of 2 times an odd number.
\[ 112 = 16 \times 7 = 2^4 \times 7 \]
Match the parts: $2^n = 16$ gives $n = 4$, and $2^{m-n} - 1 = 7$ gives $2^{m-n} = 8$, so $m - n = 3$.

Step 5: Find $m$ and $n$.
From $n = 4$ and $m - n = 3$, we get $m = 7$.

Step 6: Test each option to find the wrong one.
Option A: $m + n = 7 + 4 = 11$, true.
Option B: $2m - n = 14 - 4 = 10$, but it claims $1$, so this is false.
Option D: $3n - m = 12 - 7 = 5$, true.
The wrong statement is option (B).
\[ \boxed{2m - n = 1} \]