# help with sequence prefixes

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• May 21st 2011, 05:18 PM
Sneaky
I think my text defines it as starting from 0. In that case, sigma is {(0,0),(1,1)}. Then it is a prefix of tau according to your example. Am I wrong?
• May 21st 2011, 05:20 PM
emakarov
I would rather say, sigma, as defined above, is not a sequence.
• May 21st 2011, 05:21 PM
Sneaky
This is a summary of a definition of a sequence from my text:

A sequence of n elements taken
from set A is a function mapping from {0,1,2,...,n-1} to A. If
we call the n elements a_1,a_2,...,a_n, then the sequence is the
function that maps 0 to a_1, 1 to a_2, 2 to a_3, ..., and n-1 to
a_n. Such a function, seen as a relation, is the set of ordered
pairs {(0,a_1),(1,a_2),...,(n-1,a_n)}. E.g., the sequence <
H,E,L,L,O >, is the set {(0,H),(1,E),(2,L),(3,L),(4,O)}.

Does this change anything that you were saying?

With this in mind I still can't come up with a proper counter example, nor can I understand your example properly.
• May 21st 2011, 05:23 PM
Plato
Quote:

Originally Posted by Sneaky
Now I am confused with your previous statement with the example, so that still shows that its a subset but not a prefix? Or does there have to be a <0,0> in sigma, which then makes the example false?

I truly mean you no disrespect by this comment.
But this is an English language forum.
As such, we have very clear definitions of terms.
A sequence is a function from the positive integers to a field, real or complex.
If the sequence $\sigma_n$ is a subsequence of $\tau_n$ then $\sigma_n=\tau_{n_j}$ where $n_1.

In that westernized context, can you reframe your question?
If not, you need to find a forum that is compatible your language.
• May 21st 2011, 05:24 PM
emakarov
Quote:

Does this change anything that you were saying?
No.

It would be helpful if you posted this definition from the start. Such ubiquitous things as sequences often have slightly different definitions.
• May 21st 2011, 05:30 PM
Sneaky
OK, I'm just stuck with one thing, if you say it does not change anything you said before, and when when you say sigma is {<1,1>}, then according to the definition I posted, shouldn't it be the same as {<0,0>,<1,1>}?
• May 21st 2011, 05:34 PM
emakarov
Quote:

Originally Posted by Sneaky
With this in mind I still can't come up with a proper counter example, nor can I understand your example properly.

As I said earlier, if sequences as functions have to be defined on an initial segment of natural numbers, then being a subset is equivalent to being a prefix.
• May 21st 2011, 05:39 PM
Sneaky
So technically with that definition of sequence, there is no counter example to show that sigma is a prefix of tau but not a subset. But if sequences don't have to be defined on an initial segment of natural numbers, then your example is a valid counter example.

Is this right?
• May 21st 2011, 05:40 PM
emakarov
Yes.

Edit: In case sequences don't have to be defined on an initial segment, my example shows that being a subset does not imply being a prefix. For the other direction under the same definition, let sigma = {(1,0)} and tau = {(0,0), (1,1)}. Then sigma (representing the sequence <0>) is a prefix of tau (representing <0,1>), but is not a subset of tau.
• May 21st 2011, 06:03 PM
Sneaky
OK, thanks.
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