GATE Computer Science (CS) 2019 Solved Papers

A = {(x, X), x ∈ X and X ⊆ U}
The number of k element subsets of a set U with n elements =(

n

k

)=‌nCk
The number of possible ordered pairs (x,, X ) where x∈X is k⋅‌nCk for a given value of k from 1 to n.
So total number of ordered pairs in A So II is correct.
(Note that k = 0 is excluded since empty set has no elements and cannot form an order pair such as (x, X)). But since by the combinational identity

n

∑

k=1

k⋅(

n

k

)=n.2n−1
So I is also correct.
So both I and II are correct.