# Mathematical notation to define tuples in a set with an alphabet

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.

• Is t to appear no more than once or exactly once in each tupple? – William Elliot Oct 17 '17 at 20:26
• 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. – William Elliot Oct 17 '17 at 20:32
• If it appears in a tuple, then it appears exactly once in that tuple. Thank you for pointing that out. – Christopher A. Trotter Oct 17 '17 at 20:33
• How can you create tuples from the set building notation? That part is still unclear to me. – Christopher A. Trotter Oct 17 '17 at 20:53
• (x,y) = { {x}, {x,y} }; (x,y,z) = ((x,y),z); etc. – William Elliot Oct 18 '17 at 1:57

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 $\mid$.)
• @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. – user251257 Oct 17 '17 at 21:31