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orcarus
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How can we prove that the power set of a finite set is finite?
Thanks.
Thanks.
arildno said:Since the power set of a finite set with N elements has [itex]2^{N}[/itex] elements, set [tex]M=2^{N}+1[/tex]
Thus, there exists a natural number M so that the number of elements in the power set is less than M. But that means the power set is finite.
orcarus said:direct set-theoretical proof that doesn't require the
development of natural number exponenentiation
Background: I have seen this problem as an exercise in a number of set
theory books, but always as an exercise and never with a solution or as
a theorem with proof.
Possibly, though bear in mind that to even define 'finite' one ought to define the natural numbers. And once one has it is trivial that the (normal) power set is in bijection with the cartesian product {0,1}x{0,1}x..x{0,1} which has cardinality 2x2x2..x2, which is a finite number and has nothing to do with exponentiation.orcarus said:To clarify a little bit, I understand intuitively that if x ~ n then Px
~ 2^n and thus is obviously finite. But what I was wondering is whether
there is a direct set-theoretical proof that doesn't require the
development of natural number exponenentiation (with the associated
complexity of recursive definitions and so on).
ZioX said:You're basically doing the assuming on OP's part.
Besides, you don't need to show that a number is bounded to be finite. Any number is bounded.
You don't even need to show 2^n in the inductive step. You just need to show adding one doesn't suddenly make it infinite. Rather a trivial problem.fopc said:OP:
Let |A| = n.
If I were asked to show P(A) is finite, I'd go ahead and prove
that |P(A)| = 2^n. Clearly, 2^n is finite. So you're done.
|A| = n -> |P(A)| = 2^n is an assertion requiring proof (not a definition). There are a number of ways to prove it.
1) by Mathematical Induction. This is an instructive approach
because it causes you to think about the effect on P(A) when a single new
element is added to A, which of course is what must be done in the induction step.
The power set of a finite set is the set of all possible subsets that can be formed using the elements of the original set. It includes the empty set and the original set itself.
The power set of a finite set is represented using the notation P(S), where S is the original set. For example, if the set S = {1,2,3}, then the power set of S would be denoted as P({1,2,3}).
The cardinality (or size) of the power set of a finite set with n elements is 2^n. This means that if the original set has 3 elements, the power set will have 2^3 = 8 elements.
Yes, the power set of a finite set can be empty if the original set is also empty. The empty set is considered a subset of any set, including itself.
The power set of a finite set is always larger than the original set. This means that the number of elements in the power set is greater than the number of elements in the original set. Additionally, every element in the original set is also an element in the power set.