# Cardinality of each of these sets

my intuition tells me that C,D,E and F all the cardinality of the continuum and A and B have cardinality aleph null. Is this correct? I wouldn't know how to show it though and that's what I'm interested in. Here is the question:

Consider the following sets:

A = set of all infinite arithmetic progressions in $$\mathbb{Z}$$,

B = set of all infinite arithmetic progressions in $$\mathbb{Q}$$,

C = set of all infinite arithmetic progressions in $$\mathbb{R}$$,

D = set of all infinite sequences in $$\mathbb{R}$$,

E = set of all subsets of $$\mathbb{R}$$,

F = set of all closed balls in $$\mathbb{R^2}$$.

For each pair of these sets determine which one has larger cardinality or if they have equal cardinality.

PS: I have tried looking through the set theory section but couldn't find a similar Q and A.

• Are you aware that there are cardinalities strictly greater than the cardinality of the continuum? Have you seen the result that $x<2^x$ even for infinite cardinal numbers? – JMoravitz Apr 26 at 13:37

First, recall that for infinite sets you have $$X$$ is the same cardinality as $$X^n$$ for any positive integer $$n$$. Here I will be using $$\simeq$$ to denote "has the same cardinality as."

Arithmetic progressions can be uniquely determined by two pieces of information: the starting value and the common difference.

We see then:

• $$A\simeq \Bbb Z^2\simeq \Bbb N$$
• $$B\simeq \Bbb Q^2\simeq \Bbb N$$
• $$C\simeq \Bbb R^2\simeq \Bbb R$$

Next, the set of infinite sequences in $$\Bbb R$$ can be a little tricky, but the answers to this question show

• $$D\simeq \{f~\mid~f~:\Bbb N\times \Bbb N \to \{0,1\}\}\simeq \Bbb R$$

Now, the set of all subsets of $$\Bbb R$$ is the Power Set of $$\Bbb R$$ and by Cantor's theorem we know that the power set of any set is of strictly greater cardinality.

• $$E = \mathcal{P}(\Bbb R)$$

Finally, the set of all closed balls in $$\Bbb R^2$$ can be described using three pieces of information. The $$x$$-coordinate of the center of the ball, the $$y$$-coordinate of the center of the ball, and the radius of the ball. We see then that

• $$F \simeq \Bbb R^3\simeq \Bbb R$$

Or, if you prefer, these can be written as having cardinality $$\aleph_0,\mathfrak{c}, 2^{\mathfrak{c}}$$ respectively

• Thank you great answer. This will help me a lot. – TommySmith97 Apr 26 at 14:01
• I have one question about D: in the answers linked what does the notation {𝑓𝑛}𝑛 mean? It's been a while. – TommySmith97 Apr 26 at 14:05
• @TommySmith97 a technique called currying lets you redescribe a function which takes multiple arguments and gives an output instead as a function which takes one argument and returns a function which is then evaluated with the second argument giving the final output. It can be applied in reverse as well, a function which takes input and returns a function which later takes other input and returns a value can be thought of as a single concise function that takes both arguments as input. – JMoravitz Apr 26 at 14:10
• Ah thank you I think I understand. – TommySmith97 Apr 26 at 14:11
• @TommySmith97 as for what $\{f_n\}_n$ means, it is a sequence of functions. $(f_1,f_2,f_3,\dots)$. In this case, each function itself represents a real number and this is a sequence of real numbers, i.e. a sequence of functions. Each function is itself a function from $\Bbb N\to\{0,1\}$. – JMoravitz Apr 26 at 14:12