Aspire Faculty ID #11155 · Topic: NIMCET 2022 · Just now
NIMCET 2022

There are two sets A and B with |A| = m and |B| = n. If |P(A)| − |P(B)| = 112 then choose the wrong option (where |A| denotes the cardinality of A, and P(A) denotes the power set of A)

Solution

$|P(A)| - |P(B)| = 112$ 
$2^m - 2^n = 112$ 
$2^n(2^{m-n} - 1) = 112$ 
$112 = 2^4 \cdot 7$ 
$2^n = 2^4 \Rightarrow n = 4$ 
$2^{m-n} - 1 = 7 \Rightarrow 2^{m-n} = 8$ 
$m - n = 3 \Rightarrow m = 7$ 

$m + n = 7 + 4 = 11 ; (\text{True})$ 
$2m - n = 14 - 4 = 10 \ne 1 ; (\text{False})$ 
$2n - m = 8 - 7 = 1 ; (\text{True})$ 
$3n - m = 12 - 7 = 5 ; (\text{True})$
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