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Which of the statements regarding cardinality of sets are always correct

Let $X$ be an infinite set then

(1)$|\{F|F \subseteq X \;\text {and} \;X \; \text {is finite}\}| > $|X|$

(2) $A \in P(X) $ and $ X\setminus A $ is infinite $\rightarrow |A|<|X|$

(3) $A \subseteq X $ and $|A|<|X| \Rightarrow |X\setminus A|=|X|$

(4)$|X|<|P(X)|$

(5) $|X*X|>|X|$

and $P(X)$ indicates the power set of X. I think 4 is correct but I am not sure of the rest please provide help since we have not been covered cardinality in the course properly

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In the future, please post only one question in one post. Posting several questions in the same post is discouraged and such questions may be closed, see meta. –  Cameron Buie Jul 3 '13 at 3:52
    
For #2, consider "nice" uncountable subsets of the reals. To answer #5, we need more information (in particular, what axioms are you working with?) #3 has been answered on this site before, and I suspect that #1 has, as well. –  Cameron Buie Jul 3 '13 at 3:56
    
Are you assuming the axiom of choice? –  Andres Caicedo Jul 3 '13 at 4:00
    
@AndresCaicedo Yes I am –  user83369 Jul 3 '13 at 4:02
    
(3) is the only statement whose truth value could change in the absence of choice. (It is always true under choice, and there are models where choice fails, and there are counterexamples.) –  Andres Caicedo Jul 3 '13 at 5:49

2 Answers 2

up vote 2 down vote accepted

You are right about (4): this is Cantor’s theorem, a very important basic result about cardinalities. Assuming the axiom of choice, (5) is false: it is a general fact about infinite sets that $|X\times X|=|X|$. Unless you’re taking a course that goes into well-orderings in some detail, you could not reasonably be asked to prove this; I imagine that you’re simply expected to learn it as a fact. The same goes for (1): if $X$ is an infinite set, and $\mathscr{F}$ is the set of finite subsets of $X$, then $|\mathscr{F}|=|X|$. (3), on the other hand, is true; again, this is probably something that you’re expected simply to learn as a fact, unless you’re taking a fairly serious elementary set theory course.

Syd Henderson has given a nice, straightforward example showing that (2) is not necessarily true.

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@user14111: True, and I didn’t say otherwise: it’s the general fact that is beyond the scope of a really elementary course. –  Brian M. Scott Jul 3 '13 at 4:54

To see #2 is false, let X be the set of integers and A the set of odd integers. Then X, A and X\A all have the same cardinality.

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