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Find the cardinality of the following sets of sequences:
(a) $$\left\{(a_n)_{n \in \mathbb Q} \in \mathbb Q ^{ \mathbb N}: \forall n \in N ( a_n+a_{n+1} = a_{n+2})\right\}$$
(b) $$\left\{(a_n)_{n \in \mathbb Q} \in \mathbb Q ^{ \mathbb N}: \forall n \in N (a_n \in \mathbb Z \wedge a_n+a_{n+1} = |a_{n+2}|)\right\}$$

I have a problem with this task because I completely do not understand how I can deduce the power of this sequences from this information.
I knew that $(a_n)_{n \in \mathbb Q} \in \mathbb Q ^{ \mathbb N}$ means that sequence $a_n$ belongs to function $f$ defined as $f: \mathbb N \rightarrow \mathbb Q$ so for each $a_n$ I have the words which are measurable. What is more in both sub-points I knew three words behave in relation to each other.
Unfortunately I still do not have any knowledge how to do it. Can I get some tips?

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  • $\begingroup$ I would think an element of $\mathbb Q^{\mathbb N}$ would be a sequence of rationals (i.e. a function $\mathbb N\to\mathbb Q$). As such, the notation $(a_n)_{n\in \mathbb Q}$ does not make sense. $n$ is in $\mathbb N,$ not $\mathbb Q.$ Also, by "power" do you mean cardinality? (I also do not know what the jargon "sub-point", "word" or "measurable" mean in this context.) $\endgroup$ – spaceisdarkgreen Jan 8 at 1:34
  • $\begingroup$ Define power of set of sequences. $\endgroup$ – William Elliot Jan 8 at 4:09
  • $\begingroup$ @WilliamElliot "Power" is an old-fashioned synonym for "cardinality". $\endgroup$ – Alex Kruckman Jan 8 at 4:44
  • $\begingroup$ A sequence index with rational numbers is a strange sequence. $\endgroup$ – William Elliot Jan 8 at 9:27
  • $\begingroup$ @spaceisdarkgreen yes, you have right that this notation does not make sense, but the content is saved correctly so I am not responsible for the shortcomings of the author of the task. When it comes to my vocabulary: power is cardinality, sub-point is (a) and (b) so I talk about my two task, word is $a_{1},a_{2},...a_{n}$, measurable mean $ \in \mathbb Q$ $\endgroup$ – MP3129 Jan 8 at 17:22
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I would think an element of $\mathbb Q^{\mathbb N}$ would be a sequence of rationals (i.e. a function $\mathbb N\to\mathbb Q$). As such, the notation $(a_n)_{n\in \mathbb Q}$ does not make sense. $n$ is in $\mathbb N,$ not $\mathbb Q.$ Also, by "power" do you mean cardinality? (I also do not know what the jargon "sub-point", "word" or "measurable" mean in this context.)

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