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I am a computer science student trying to wrap my head around some mathematical notation, so is there any elegant mathematical notation for the following:

I would like to define a $Set$, denoted $K^{n}$, such that the elements in the $Set$ are tuples, $a_{1}, ..., a_{n}$, where $a_{i} \in \{0, t, 1\}, 1 \le i \le n$. As shown by the example below:

Example:

$K^{3}$ = $\{<0,0,0>,<t,0,0>,<1,0,0>,<1,t,0>,... \}$

Also, are there any notation for describing multiple constraints of what kind of elements the $Set$ can contain?

Example:

Constraint 1: If $X \in K^{n}$, then $\exists!a_{i}=\{t\}, 1 \le i \le n$. (There can only be one unique $\{t\}$ in $a_{1},...,a_{n}$)

$<0,t,t> \notin K_{3}$.

Apologize for my terrible notation.

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  • $\begingroup$ Is t to appear no more than once or exactly once in each tupple? $\endgroup$ – William Elliot Oct 17 '17 at 20:26
  • $\begingroup$ Set building notation is { x : P(x) } for the set of all x such that P(x) where P is some statement about x which could be a conjunction of statements about x. $\endgroup$ – William Elliot Oct 17 '17 at 20:32
  • $\begingroup$ If it appears in a tuple, then it appears exactly once in that tuple. Thank you for pointing that out. $\endgroup$ – Christopher A. Trotter Oct 17 '17 at 20:33
  • $\begingroup$ How can you create tuples from the set building notation? That part is still unclear to me. $\endgroup$ – Christopher A. Trotter Oct 17 '17 at 20:53
  • $\begingroup$ (x,y) = { {x}, {x,y} }; (x,y,z) = ((x,y),z); etc. $\endgroup$ – William Elliot Oct 18 '17 at 1:57
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You basically have it.

Say $K=\{ 0, t, 1 \}$ is the set of possible values in these tuples. Then, $K^n$ is the set of $K$-valued $n$-tuples.

If I understand your question correctly, you asked for set of tuples containing $t$ exactly once: $$ S = \{ a\in K^n \mid \exists! i \text{ such } 1\le i \le n \text{ and } a_i = t \}. $$ (It is pretty valid to use normal English in the condition part after $\mid$.)

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  • $\begingroup$ If one was to add multiple conditions besides the one given in the example, is there a nice way to do this? That notation clarified the confusion I had. Thank you. $\endgroup$ – Christopher A. Trotter Oct 17 '17 at 21:16
  • $\begingroup$ @ChristopherA.Trotter the usual logical connectives are “and” ($\land$), “or” ($\lor$) and sometimes “implication” ($\implies$), and “equivalence” ($\iff$). You can also build different sets for different conditions and connect them by set operations, which are essential the same. $\endgroup$ – user251257 Oct 17 '17 at 21:31

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