(1)
Set A={ Vincenzo Yasmeen Zipei };
P(A)={ { } { Vincenzo } { Yasmeen } { Zipei } { Vincenzo Yasmeen } { Vincenzo Zipei } { Yasmeen Zipei } { Vincenzo Yasmeen Zipei } }. Note that the subsets include the empty set and the set A itself.
(2)
B ∪ { Zipei } = { Vincenzo Yasmeen Zipei };
P(B ∪ { Zipei })={ { } { Vincenzo } { Yasmeen } { Zipei } { Vincenzo Yasmeen } { Vincenzo Zipei } { Yasmeen Zipei } { Vincenzo Yasmeen Zipei } }.
C∩P(B)={ } by definition of disjoint.
P(B) = { { } { Vincenzo } { Yasmeen } { Vincenzo Yasmeen } } (P(B) has 4 elements)
P(B)∪C={ { } { Vincenzo } { Yasmeen } { Vincenzo Yasmeen } {C} }
Therefore, C∩P(B)={ }=C∩{ { } { Vincenzo } { Yasmeen } { Vincenzo Yasmeen } }.
C could be { } so that C∩P(B)={ } and P(B)∪C={ { } { Vincenzo } { Yasmeen } { Vincenzo Yasmeen } } = P(B). But this implies C has only one element.
If C={ 1 2 3 4 }, for example, P(B)∪C={ { } { Vincenzo } { Yasmeen } { Vincenzo Yasmeen } { 1 2 3 4 } } then C∩P(B)={ 1 2 3 4 }∩{ { } { Vincenzo } { Yasmeen } { Vincenzo Yasmeen } }={ }.
(3)
Pairing: (1,{ }), (2,{ Vincenzo }), (3,{ Yasmeen }, (4,{ Vincenzo })
(4)
The number of elements in a power set A = 2ⁿ where n=|A|. If A has three elements, P(A) has 2³=8 elements.
When A={ 1 ... 2020 ), |A|=2020, |P(A)|=2²⁰²⁰. |P({2021}∪A)|=2²⁰²¹, which is 2|P(A)|.
(5) 2.4078×10⁶⁰⁸ approximately.