How to write a math notation for a set generated by a relation?

Question

I am trying to write a mathematical notation for a set of variables and their values over time.

A function $f$ should map each of $N$ variables to a value of $0$ or $1$ over time and the results are stored in a set $S$ in an accumulative manner.

For example:

• $N = 3$
• variables: $a, b, c ∈ [0, 5]$
• time($t$): $0...3$

 time      f(a),f(b),f(c)              S
======    ===============      ===================
  0            0,0,0           {(a,0),(b,0),(c,0)}
  1            0,1,0           {(a,0),(b,1),(c,0)}
  2            1,0,0           {(a,1),(b,1),(c,0)}
  3            1,0,1           {(a,1),(b,1),(c,1)}

How can I write a mathematical notation for set $S$?

Answer 1

Possibly you could use state notation as I proposed at this question, where your state = time.

Answer 2

One way to think of your example is as a sequence of functions $f_1, f_2, \ldots$ each of which maps $N$ to $\{0,1\}$. Then when you think of a function as a set of ordered pairs, the sets you want are just the functions themselves.

Thinking of a function as a set of ordered pairs is usually a formality that stays in the background in most applications. Here it is exactly what you want. Perhaps you can just say that at the start of your document.

In your example, the value of $f$ changes from $0$ to $1$ at just one point at each (time) step. The inverse image of $1$ increases by $1$ at each step, reflecting the sense in which something is "growing".

If you are always going to have $|N|$ steps then your sequence of functions is easily encoded as a permutation of $N$, indicating the order in which its elements are "added".