Sets - University

Question

Given a set A, we define its power set, P(A), to be the set of all subsets of A: P(A) = {B: B⊆A}.

(1)  Let A = {Vincenzo, Yasmeen, Zipei}.  Find P(A);

(2)  Let B = {Vincenzo,Yasmeen}.  Find the set C such that P( B ∪ {Zipei}) = P(B)∪C, and C is disjointfrom P(B).

(3)  Continuing from part 2, show that C and P(B) have the same number of elements by pairing each element of P(B) with an element of C in a “natural” way.

(4)  Let A be the finite set {1,2, ...,2020}.  Show that P(A∪ {2021}}) has twice as many elements as P(A).

(5)  Find, with proof, the number of elements of P({1,2, . . . ,2021}).  Your answer should be a number.  (Note: We are not expecting you to use induction for this question.)

Answer 1

(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.

 

Comments

thank you so much!!