Bijection between ${0,1}^*$ and the natural numbers.
$begingroup$
So the tasks is to show that ${0,1}^*$ is countable.
So the idea that i am having is that each number can be mapped to it's own in decimal.
$f(1001)= 9$
$f(101)=5$
But what happens with all the string which start with zeros.
For example: $f(01001)=$? $f(001001)=$? $f(00000000001001)=$?.
Is my idea completely wrong? Any tips?
elementary-set-theory formal-languages natural-numbers
asked Dec 29 '18 at 14:11
Angeld55Angeld55
565
$endgroup$
add a comment |
$begingroup$
So the tasks is to show that ${0,1}^*$ is countable.
So the idea that i am having is that each number can be mapped to it's own in decimal.
$f(1001)= 9$
$f(101)=5$
But what happens with all the string which start with zeros.
For example: $f(01001)=$? $f(001001)=$? $f(00000000001001)=$?.
Is my idea completely wrong? Any tips?
elementary-set-theory formal-languages natural-numbers
asked Dec 29 '18 at 14:11
Angeld55Angeld55
565
$endgroup$
1
$begingroup$
What exactly is ${0,1}^{*}$ ?
$endgroup$
– Peter Melech
Dec 29 '18 at 14:14
$begingroup$
The set of all strings with the symbols 0 and 1
$endgroup$
– Angeld55
Dec 29 '18 at 14:16
2
$begingroup$
en.wikipedia.org/wiki/Bijective_numeration
$endgroup$
– r.e.s.
Dec 29 '18 at 14:17
add a comment |
$begingroup$
So the tasks is to show that ${0,1}^*$ is countable.
So the idea that i am having is that each number can be mapped to it's own in decimal.
$f(1001)= 9$
$f(101)=5$
But what happens with all the string which start with zeros.
For example: $f(01001)=$? $f(001001)=$? $f(00000000001001)=$?.
Is my idea completely wrong? Any tips?
elementary-set-theory formal-languages natural-numbers
asked Dec 29 '18 at 14:11
Angeld55Angeld55
565
$endgroup$
So the tasks is to show that ${0,1}^*$ is countable.
So the idea that i am having is that each number can be mapped to it's own in decimal.
$f(1001)= 9$
$f(101)=5$
But what happens with all the string which start with zeros.
For example: $f(01001)=$? $f(001001)=$? $f(00000000001001)=$?.
Is my idea completely wrong? Any tips?
elementary-set-theory formal-languages natural-numbers
elementary-set-theory formal-languages natural-numbers
asked Dec 29 '18 at 14:11
Angeld55Angeld55
565
asked Dec 29 '18 at 14:11
Angeld55Angeld55
565
edited Dec 29 '18 at 14:15
Angeld55
asked Dec 29 '18 at 14:11
Angeld55Angeld55
565
asked Dec 29 '18 at 14:11
Angeld55Angeld55
565
asked Dec 29 '18 at 14:11
Angeld55Angeld55
565
565
1
$begingroup$
What exactly is ${0,1}^{*}$ ?
$endgroup$
– Peter Melech
Dec 29 '18 at 14:14
$begingroup$
The set of all strings with the symbols 0 and 1
$endgroup$
– Angeld55
Dec 29 '18 at 14:16
2
$begingroup$
en.wikipedia.org/wiki/Bijective_numeration
$endgroup$
– r.e.s.
Dec 29 '18 at 14:17
add a comment |
1
$begingroup$
What exactly is ${0,1}^{*}$ ?
$endgroup$
– Peter Melech
Dec 29 '18 at 14:14
$begingroup$
The set of all strings with the symbols 0 and 1
$endgroup$
– Angeld55
Dec 29 '18 at 14:16
2
$begingroup$
en.wikipedia.org/wiki/Bijective_numeration
$endgroup$
– r.e.s.
Dec 29 '18 at 14:17
1
1
$begingroup$
What exactly is ${0,1}^{*}$ ?
$endgroup$
– Peter Melech
Dec 29 '18 at 14:14
$begingroup$
What exactly is ${0,1}^{*}$ ?
$endgroup$
– Peter Melech
Dec 29 '18 at 14:14
$begingroup$
The set of all strings with the symbols 0 and 1
$endgroup$
– Angeld55
Dec 29 '18 at 14:16
$begingroup$
The set of all strings with the symbols 0 and 1
$endgroup$
– Angeld55
Dec 29 '18 at 14:16
2
2
$begingroup$
en.wikipedia.org/wiki/Bijective_numeration
$endgroup$
– r.e.s.
Dec 29 '18 at 14:17
$begingroup$
en.wikipedia.org/wiki/Bijective_numeration
$endgroup$
– r.e.s.
Dec 29 '18 at 14:17
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Hint: Consider the map ${0,1}^* to mathbb N$ given by $w mapsto (1w)_2$.
This assumes that $0 notin mathbb N$. If $0 in mathbb N$, use $w mapsto (1w)_2-1$.
answered Dec 29 '18 at 14:17
lhflhf
168k11172404
$endgroup$
add a comment |
$begingroup$
You can do this trick. Instead of sending each sequence in ${0,1}^star$ to it's decimal try to first add the digit $1$ at the beginning and then send it to it's decimal (i.e. if you begin with 01 you change it to 101 which is 5 in decimal). That's how you will overcome the zeros problem.
answered Dec 29 '18 at 14:48
YankoYanko
8,4692830
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add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
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active
oldest
votes
$begingroup$
Hint: Consider the map ${0,1}^* to mathbb N$ given by $w mapsto (1w)_2$.
This assumes that $0 notin mathbb N$. If $0 in mathbb N$, use $w mapsto (1w)_2-1$.
answered Dec 29 '18 at 14:17
lhflhf
168k11172404
$endgroup$
add a comment |
$begingroup$
Hint: Consider the map ${0,1}^* to mathbb N$ given by $w mapsto (1w)_2$.
This assumes that $0 notin mathbb N$. If $0 in mathbb N$, use $w mapsto (1w)_2-1$.
answered Dec 29 '18 at 14:17
lhflhf
168k11172404
$endgroup$
add a comment |
$begingroup$
Hint: Consider the map ${0,1}^* to mathbb N$ given by $w mapsto (1w)_2$.
This assumes that $0 notin mathbb N$. If $0 in mathbb N$, use $w mapsto (1w)_2-1$.
answered Dec 29 '18 at 14:17
lhflhf
168k11172404
$endgroup$
Hint: Consider the map ${0,1}^* to mathbb N$ given by $w mapsto (1w)_2$.
This assumes that $0 notin mathbb N$. If $0 in mathbb N$, use $w mapsto (1w)_2-1$.
answered Dec 29 '18 at 14:17
lhflhf
168k11172404
answered Dec 29 '18 at 14:17
lhflhf
168k11172404
answered Dec 29 '18 at 14:17
lhflhf
168k11172404
answered Dec 29 '18 at 14:17
lhflhf
168k11172404
168k11172404
add a comment |
add a comment |
$begingroup$
You can do this trick. Instead of sending each sequence in ${0,1}^star$ to it's decimal try to first add the digit $1$ at the beginning and then send it to it's decimal (i.e. if you begin with 01 you change it to 101 which is 5 in decimal). That's how you will overcome the zeros problem.
answered Dec 29 '18 at 14:48
YankoYanko
8,4692830
$endgroup$
add a comment |
$begingroup$
You can do this trick. Instead of sending each sequence in ${0,1}^star$ to it's decimal try to first add the digit $1$ at the beginning and then send it to it's decimal (i.e. if you begin with 01 you change it to 101 which is 5 in decimal). That's how you will overcome the zeros problem.
answered Dec 29 '18 at 14:48
YankoYanko
8,4692830
$endgroup$
add a comment |
$begingroup$
You can do this trick. Instead of sending each sequence in ${0,1}^star$ to it's decimal try to first add the digit $1$ at the beginning and then send it to it's decimal (i.e. if you begin with 01 you change it to 101 which is 5 in decimal). That's how you will overcome the zeros problem.
answered Dec 29 '18 at 14:48
YankoYanko
8,4692830
$endgroup$
You can do this trick. Instead of sending each sequence in ${0,1}^star$ to it's decimal try to first add the digit $1$ at the beginning and then send it to it's decimal (i.e. if you begin with 01 you change it to 101 which is 5 in decimal). That's how you will overcome the zeros problem.
answered Dec 29 '18 at 14:48
YankoYanko
8,4692830
answered Dec 29 '18 at 14:48
YankoYanko
8,4692830
answered Dec 29 '18 at 14:48
YankoYanko
8,4692830
answered Dec 29 '18 at 14:48
YankoYanko
8,4692830
8,4692830
add a comment |
add a comment |
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1
$begingroup$
What exactly is ${0,1}^{*}$ ?
$endgroup$
– Peter Melech
Dec 29 '18 at 14:14
$begingroup$
The set of all strings with the symbols 0 and 1
$endgroup$
– Angeld55
Dec 29 '18 at 14:16
2
$begingroup$
en.wikipedia.org/wiki/Bijective_numeration
$endgroup$
– r.e.s.
Dec 29 '18 at 14:17