mod function and maps

[ifoan]

I have two problems I need to complete for my discrete structures class. I've finished all the rest except these two. Can anyone help me out? I'm a CIS student and don't have an extremly strong backgroun in math.

I do not know where to start with this one:
There exist, for each element s in Z, unique elements u in Z and \(\displaystyle \mu_n(s)\) in \(\displaystyle Z_n\) such that \(\displaystyle s=nu+\mu_n (s)\) Thus,

\(\displaystyle \mu_n : Z \rightarrow Z_n, z \mapsto \mu_n(z)\)

is a map from \(\displaystyle Z\) to \(\displaystyle Z_n\). This map is called a mod function for n.

Write down all the values of the map

\(\displaystyle \sigma : Z_{15} \rightarrow Z_{3} \times Z_{5}, r \rightarrow (\mu_{3}(r),\mu_{5}(r))\)

(thereby showing that \(\displaystyle \sigma\) is bijective.


Any help is really appreciated!

JJ

Someone has told me this but im trying to still figure it out:

Each element of the map \(\displaystyle \sigma\) has the form

\(\displaystyle a \rightarrow (b,c)\)

where a is a member of \(\displaystyle Z_{15}\), b is a member of \(\displaystyle Z_3\) and c is a member of \(\displaystyle Z_5\).

For example, if \(\displaystyle \mu_{15}(n)=7\) (i.e. n=7 mod 15), then \(\displaystyle \mu_{3}(n)=1\) and \(\displaystyle \mu_{5}(n)=2\), so one element of \(\displaystyle \sigma\) is:

\(\displaystyle 7 \rightarrow (1,2)\)

Go through all 15 members of \(\displaystyle Z_{15}\) and show that each one maps to a different member of \(\displaystyle Z_3 \times Z_5\).

would
\(\displaystyle 6 \rightarrow (0,1)\)
\(\displaystyle 5 \rightarrow (1,4)\)
\(\displaystyle 4 \rightarrow (3,1)\)

be elements also? Im trying to figure out how he got those elements and how to get the rest

 

[pka]

I am not sure of the notation your text/instructor is using.

But in the above graphic, the first column is Z<SUB>15</SUB>.
The second column is the mapping Z<SUB>15</SUB> to Z<SUB>3</SUB>.
The third column is the mapping Z<SUB>15</SUB> to Z<SUB>5</SUB>.
I hope this helps you.

 

[ifoan]

that helps me tremendously, becaue its the answer...but i still would like to know how you did that?

 

[pka]

Actually it was done with a scaled-down version of MathCad.
That is a computer algebra system.

 

[ifoan]

oh ok, i should try to get a trail version of that to see how it works

thank you very much!

 

[marcmtlca]

modular arithmetic

Hi, so basically what is happening here is division. No math programs are necessary.

take any integer \(\displaystyle n \in \bf{N}\). Then consider dividing that number by some other number k. You then express your answer as some multiplier of k plus some remainder r.

Example: Divide 25 by 8.

Answer: 25=8*3+1

So the way that this relates to your problem is if you choose \(\displaystyle Z_{8}\). The unique map that the question talks about will give \(\displaystyle \mu_{n}(s)=1\) and \(\displaystyle u=3\).

So now we can look at what is going on when you take a number in \(\displaystyle Z_{15}\) and divide it by 5 in the same way.

looking at the number 1: you get 0*5+1
looking at the number 7: you get 1*5+2
looking at the number 13: you get 2*5+3

So, the question is asking for a map from \(\displaystyle Z_{15}\) to the cartesian product \(\displaystyle Z_{3} \times Z_{5}\)

and so:
0 -> (0,0)
1 -> (0,1)
2 -> (0,2)
5 -> (1,0)
13 -> (2,3)

which is slightly different from what pka found using his program.

 

[ifoan]

thank you everyone!

pka, would you be able to send me the file that you used to create this? the mcd file. im trying to learn how to use it

 

[marcmtlca]

Do not use a computer program for this. There are two reasons.

1: your answer will be incorrect
2: You will not understand what is going on really.

 

[pka]

Do not use a computer program for this. There are two reasons.
1: your answer will be incorrect
2: You will not understand what is going on really.

That is nonsense.
I read your response to this question and the way you read it is wrong!
Your statement “1: your answer will be incorrect” means that I am wrong.
But I know this problem; I have taught this problem; you are the one just wrong.
Of course, you are welcome to post here.
But please stay with topics you really understand.

 

[ifoan]

i feel there is nothing wrong with using a computer program to corrobrate your answers.

i work them out, then try to see if im right.

pka, i will send you a pm!

 

[marcmtlca]

Hmm... you were right actually, sorry.

Although I stand by my stance on computer programs.