Cardinality and Power set?

Question

These are some confusing questions I came across. What are the answers and is there any rule that can be used when there are nested sets as shown below?

What is the power set of {{},{{}}}?
What is the cardinality of {a,{a}}?
What is the cardinality of {{a}}?
What is the cardinality of {a,{a},{a,{a}}}?

Answer 1

Given an n-element set, the cardinality of its power set is 2^n; with this, you should be able to solve the last three questions; for example, in the third question, you have the set A = {{a}}, which has only one element: {a}; therefore, its power set must have 2^1 = 2 elements, and these must be the empty set (its a subset of any set), and the set A himself (because A is always a subset of himself); if you have a set with more than an element, then its proper subsets must be included too.

Regarding the power of the first question's set, it's helpful to change the notation a little to:

{{},{{}}} = {∅ {∅}}

Which is somewhat less messy; now its easier to see that this is a set with only two elements, which means that its power must have 4 elements, and these are:

∅, {∅}, {{∅}}, {∅,{∅}}

As for an algorithm, you may try this if the cardinality of your set A is not too high: list its elements in a row, and below the 2^n binary sequences; now there is a bijection between these and the subsets of A: take a sequence and form the corresponding subset by selecting the elements corresponding to the 1's in the sequence; try with the example above.

Answer 2

purely record all subsets. for instance {Earth, solar} should be one subset. The set whose aspects are each and each and every of the subsets is the potential set. there'll be 2^3 = 8 aspects interior the potential set. do not overlook the empty set is a subset of each and every set, and cardinality for finite units is purely the count number of ways many aspects are interior the set. also, for instance, the set {{Earth,Moon}, {Earth,solar}} has 2 aspects no longer 3. those 2 aspects are: {Earth,Moon} and {Earth,solar}.

Answer 3

http://en.wikipedia.org/wiki/Cardinality