How can I express that a n-tuple contains an element at least once?

Question

This is a very simple question, yet I could not find a satisfying answer for it.

Consider the set $S =\{a, b, c\}$. To describe the fact that the set $S$ contains $b$, you can write $b \in S$.

But what about the triple $t = (a, b, c)$? How do you describe that this triple contains $b$ at least once? Is it sound to write $b \in t$, as long as I define the meaning?

Answer 1

It depends on the definition of tuple. Actually a tuple as a concept needs only to be able to respond to specific coordinates questions (such as "is $b$ the second element of $t$?").

In the classical set representation of pairs, $(a, b) = \{\{a\}, \{a,b\}\}$ so the predicate $t\in\bigcup t$ is equivalent to $b$ being either the first or second element of the pair.

Answer 2

$t \in (S\times S \times S) \backslash (\{a,c\}\times\{a,c\}\times\{a,c\})$ I guess.