UPSC CDS Math Number System Questions

Number of non-empty subsets of A=2m−1
Number of non-empty subsets of B=2n−1
∴2m−1=2k+2n−1
⇒2m=2k+2n
When m=n+2
2n+2=2k+2n⇒2n(4−1)=2k
⇒3=2k−n,which is not possible as LHS is odd and RHS is even.
When m=n+1
2n+1=2k+2n⇒2n(2−1)=2k
⇒2n=2k,which is possible when n = k.
When m=n+p
2n+p=2k+2n⇒2n(2p−1)=2k
⇒2p−1=2k−n,which is not possible as LHS is odd and RHS is even
When m=n+t
2n+t=2k+2n⇒2n(2t−1)=2k
⇒2t−1=2k−n,which is not possible as LHS is odd and RHS is even.
Trick :
Take m=2,n=2
Then,23−1=7=22+(22−1)
⇒m=n+1