Question

Suppose is a subset of the cartesian product of a finite set and uncountable set, anuBis the product two countably infinite sets.What can you say about|A∩B|? What can you say about|P(A∪B)|?

Answer 1

Case 1:

Let A= R × { 1 } i.e. A is Cartesian product of an uncountable set R with a finite set { 1 }. Then A is also an uncountable set. Now we again let B = N × N ,where N is set of natural number therefore B is countably infinite set. Thus here we find that

A $\cap$ B = N ×{ 1} which is countably infinite set more clear it is similar to N ,set of natural numbers. Therefore cardinality of   A $\cap$ B is same as cardinality of N means countably infinite. Also ( A U B)= R × { 1 } which is similar to R therefore it's cardinality will be same as R or power set of natural number ,i .e. uncountable set.Hence

| A $\cap$ B | = Aliph naught {countably infinite set }

|P(A $\cup$ B)| = c ,where c is continum which denotes cardinality of P(N).

Case2:

Let A = Q^{​​​​​​c} × { 1} ,i.e. A is product of an uncountable set with a countable set and also let B =N × N, i.e.

B is producproductwo countably infinite set. Therefore we get (A$\cap$ B ) is empty set and ( A U B ) is again uncountable set whoes cardinality is similar to power set of Natural numbers P(N) i. e.

|A $\cap$ B| = 0.

|P(A U B)| = c ,where c is continum which denotes the cardinality of P(N).

Thus from above two cases we can conclude that the set A$\cap$​​​​​​​ B is either countably infinite or countably finite but power set of ( A U B ) is always uncountable set