Step-by-step Solution:

\[ \textbf{Given: } |A| = m,\quad |B| = n,\quad |P(A)| - |P(B)| = 112 \] \[ \text{We know that } |P(A)| = 2^m,\quad |P(B)| = 2^n \] \[ \Rightarrow 2^m - 2^n = 112 \] \[ \textbf{Step 1: Factorize} \] \[ 2^m - 2^n = 2^n\left(2^{m-n} - 1\right) = 112 \] \[ 112 = 16 \times 7 = 2^4 \times 7 \] \[ \textbf{Step 2: Compare factors} \] Since \(2^{m-n} - 1\) is odd, it must be equal to \(7\). \[ \Rightarrow 2^{m-n} - 1 = 7 \Rightarrow 2^{m-n} = 8 \Rightarrow m - n = 3 \] \[ \text{Also, } 2^n = 16 \Rightarrow n = 4 \] \[ \Rightarrow m = n + 3 = 7 \] \[ \textbf{Step 3: Check options} \] \[ \text{A: } m + n = 7 + 4 = 11 \quad (\text{True}) \] \[ \text{B: } 2m - n = 2(7) - 4 = 14 - 4 = 10 \ne 1 \quad (\text{False}) \] \[ \text{C: } 2n - m = 2(4) - 7 = 8 - 7 = 1 \quad (\text{True}) \] \[ \text{D: } 3n - m = 3(4) - 7 = 12 - 7 = 5 \quad (\text{True}) \] \[ \therefore \quad \text{Wrong option is B} \] \[ \boxed{\text{Option B}} \]