Mixing
Find the cardinality of the following sets of sequences
$begingroup$
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?
elementary-set-theory logic
asked Jan 7 at 22:42
MP3129MP3129
2237
$endgroup$
add a comment |
$begingroup$
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?
elementary-set-theory logic
asked Jan 7 at 22:42
MP3129MP3129
2237
$endgroup$
$begingroup$
I would think an element of $mathbb Q^{mathbb N}$ would be a sequence of rationals (i.e. a function $mathbb Ntomathbb Q$). As such, the notation $(a_n)_{nin mathbb Q}$ 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
add a comment |
$begingroup$
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?
elementary-set-theory logic
asked Jan 7 at 22:42
MP3129MP3129
2237
$endgroup$
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?
elementary-set-theory logic
elementary-set-theory logic
asked Jan 7 at 22:42
MP3129MP3129
2237
asked Jan 7 at 22:42
MP3129MP3129
2237
edited Jan 8 at 20:36
MP3129
asked Jan 7 at 22:42
MP3129MP3129
2237
asked Jan 7 at 22:42
MP3129MP3129
2237
asked Jan 7 at 22:42
MP3129MP3129
2237
2237
$begingroup$
I would think an element of $mathbb Q^{mathbb N}$ would be a sequence of rationals (i.e. a function $mathbb Ntomathbb Q$). As such, the notation $(a_n)_{nin mathbb Q}$ 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
add a comment |
$begingroup$
I would think an element of $mathbb Q^{mathbb N}$ would be a sequence of rationals (i.e. a function $mathbb Ntomathbb Q$). As such, the notation $(a_n)_{nin mathbb Q}$ 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
$begingroup$
I would think an element of $mathbb Q^{mathbb N}$ would be a sequence of rationals (i.e. a function $mathbb Ntomathbb Q$). As such, the notation $(a_n)_{nin mathbb Q}$ 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
$begingroup$
I would think an element of $mathbb Q^{mathbb N}$ would be a sequence of rationals (i.e. a function $mathbb Ntomathbb Q$). As such, the notation $(a_n)_{nin mathbb Q}$ 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
$begingroup$
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
$begingroup$
@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
$begingroup$
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
$begingroup$
@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
add a comment |
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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 $mathbb Ntomathbb Q$). As such, the notation $(a_n)_{nin mathbb Q}$ 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