# Statements on Cardinality of sets

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)$indicates the power set of X. I think 4 is correct but I am not sure of the rest please provide help since we have not been covered cardinality in the course properly - In the future, please post only one question in one post. Posting several questions in the same post is discouraged and such questions may be closed, see meta. – Cameron Buie Jul 3 '13 at 3:52 For #2, consider "nice" uncountable subsets of the reals. To answer #5, we need more information (in particular, what axioms are you working with?) #3 has been answered on this site before, and I suspect that #1 has, as well. – Cameron Buie Jul 3 '13 at 3:56 Are you assuming the axiom of choice? – Andres Caicedo Jul 3 '13 at 4:00 @AndresCaicedo Yes I am – user83369 Jul 3 '13 at 4:02 (3) is the only statement whose truth value could change in the absence of choice. (It is always true under choice, and there are models where choice fails, and there are counterexamples.) – Andres Caicedo Jul 3 '13 at 5:49 ## 2 Answers You are right about (4): this is Cantor’s theorem, a very important basic result about cardinalities. Assuming the axiom of choice, (5) is false: it is a general fact about infinite sets that$|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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