Understanding nX+M=y Sequence M_i

[honestrosewater]

(keeping in mind some conventional symbols are here written out in words...)

For

n * X + M = y

where

n is in N,
X is in Z and nonnegative,
M is in Z and nonnegative and less than n,
and y is, of course, defined by n, X, and M

sequence
M_i = (n * X + i) [X = (0, 1, 2, ...)]

where i takes, in turn, each value of M for some n (or for each n, or when n is constant, how should I say this?). So when n = 3, then M = {0, 1, 2} and M_i denotes collectively the sequences

M_0 = ( n * X + 0 ) [X = ( 0, 1, 2, ... )]
M_1 = ( n * X + 1 ) [X = ( 0, 1, 2, ... )]
M_2 = ( n * X + 2 ) [X = ( 0, 1, 2, ... )],

i.e, there is a sequence for each value, i, that M takes.

Does a better way of saying this jump out at anyone?

Any help will be greatly appreciated, as I am working on my own and don't know how this is conventionally expressed.

Rachel

 

[HallsofIvy]

Perhaps just M_i = {(n * X + i) [X = (0, 1, 2, ...)]} [i= 0, 1, 2, ...] is what you are seeking. It is understood, of course, that n is a constant.

 

[honestrosewater]

Whew, I'm so relieved that you understood what I was saying.

As stated:

M_i = {(n * X + i) [X = (0, 1, 2, ...)]} [i= 0, 1, 2, ...]

would I not have to specify [i = 0, 1, 2, ..., (n-1)]

or would adding (n-1) just be confusing/superfluous since I have already defined M as a nonegative integer less than n?

Also,
I will later be defining sets for each sequence of M_i,

Let M_j be the set of all t such that t is in M_j if t is a term in sequence M_i, where j = i as i runs through M.
(That is, there is a set M_j for EACH sequence M_i, where i takes different values. Set M_j is NOT the union of all the sequences collectively denoted by M_i. If there are n sequences, there are n sets.)

Which of these would be preferable:

Let set M_j = {t : t is in M_i, i = j}

Let set M_j = {e : e = t for some t in M_i, i = j}

Let set M_j = {e : e = t_x for some t_x in M_i, i = j}

etc... Should I enclose M_i in some type of bracket, ex. (M_i), to show it is a sequence? Should I include the formula for M_i?

Or to answer all of these questions, as long is my meaning is clear, can I not be so anal?

BTW, either way is fine, I'd just rather pick one and stick with it.

Many thanks and happy thoughts
Rachel