What is a tuple? In theory of computation, DFA's, NFA's, etc. are represented as a "tuple". Probability spaces are tuples. I am confused on what the notion of a tuple is and how it differs from a set?
 A: @Henry has given an explanation from a general view. Now I give an explanation in mathematical logic (set theory) with a rigorous way which could be seen as a mathematical translation of Henry's words. I define tuples in my book as follows.
Definition 1 (Tuple). Let $n\in\mathbb{N}$ and $x_0.\cdots,x_{n-1}$ be sets. Set
\begin{align*}
  (\,)&=\varnothing,\\
  (x_0)&=x_0,\\
(x_0,x_1)&=\{\{x_0\},\{x_0,x_1\}\},\\
\vdots\qquad&\qquad\qquad~\vdots\\
(x_0,\cdots,x_{n-1})&=((x_0,\cdots,x_{n-2}),x_{n-1}).
\end{align*}
$(x_0,\cdots,x_{n-1})$ is called an ordered $n$-tuple or $n$-tuple, and also written as $(x_i\mid i<n)$ or $(x_i)_{i<n}$.
Lemma 2. Let $m,n\in\mathbb{N}$, $(x_i\mid i<m)$ be an $m$-tuple and $(y_i\mid i<n)$ be an $n$-tuple. Then $(x_i\mid i<m)=(y_i\mid i<n)$ if and only if $m=n$ and $x_i=y_i$ for all $i<m$.
Remark 3. (1) Ordered $2$-tuples are also called ordered pairs or couples, and for alternative names for other $n$-tuples of specific lengths, please see the link.
(2) By Lemma 2, $n$-tuples have orders following from $2$-tuple defined by K. Kuratowski.
(3) $n$-tuples are finite.
(4) For more remark on Definition 1, see my another post.
Furthermore, the concept of tuples is related to that of sequences which are defined in my book as follows.
Definition 4 (Sequence). Let $s$ be a set and $\alpha$ be an ordinal.
(1) $s$ is a sequence if $s$ is a function with some ordinal as the domain, i.e., $s$ is a function on some ordinal; we call $s$ is an $\alpha$-sequence if its domain is $\alpha$; and we may also write $s$ as $\langle s_\xi\mid \xi<\alpha\rangle$ or $\langle s_\xi\rangle_{\xi<\alpha}$ if $s(\xi)=s_\xi$ for all $\xi<\alpha$.
(2) Suppose $s$ is an $\alpha$-sequence. We call $s_\xi$ is the $\xi$-th value of $s$ which is written as $s(\xi)=s_\xi$; and we call $\alpha$ is the length of $s$ which is written as $\mathrm{len}(s)=\alpha$.
(3) Suppose $s$ is an $\alpha$-sequence. Clearly, $s$ is empty if and only if $\alpha=0$; $s$ is finite if and only if there is some $n\in\mathbb{N}$ such that $\alpha=n$, and at the moment it may be written as $\langle s_0,\cdots,s_{n-1}\rangle$, or simply as $s_0\cdots s_{n-1}$ considering of readability; $s$ is infinite if and only if $\alpha\geq\omega$.
Lemma 5. Let $\alpha,\beta$ be ordinals, $\langle s_\xi\mid\xi<\alpha\rangle$ be an $\alpha$-sequence and $\langle t_\xi\mid\xi<\beta\rangle$ be a $\beta$-sequence. Then $\langle s_\xi\mid\xi<\alpha\rangle=\langle t_\xi\mid\xi<\beta\rangle$ if and only if $\alpha=\beta$ and $s_\xi=t_\xi$ for all $\xi<\alpha$.
Remark 6. (1) By Lemma 5, sequences also have orders following from ordinals.
(2) Sequences may be infinite.
(3) $(x_0,\cdots,x_{n-1})$ and $\langle x_0,\cdots,x_{n-1}\rangle$ are almost the same in appearance except the parenthesis or angle brakets, and so we can regard that tuples are finite sequences, or sequences are generalizations of tuples. Hence someone may also define $n$-tuples as $n$-sequence for $n\neq 1,2$ (Why?).
(4) But note that $n$-tuples and $n$-sequence are not equal even if their elements are equal one by one, for example, $(3,4)\neq \{(0,3),(1,4)\}=\langle 3,4\rangle$.
(5) For more remark on Definition 4, see my another post.
Hope these useful for you!
A: In the context you are asking about, a tuple is just a (short, finite) list of the objects that you need to specify a particular mathematical structure.
For example, for a vector space you need
$$
(V, F, +, \cdot)
$$
where $V$ is a set, $F$ is a field and $+$ and $\cdot$ are the functions that tell you vector addition and scalar multiplication. Then you state the axioms.
For a DFA the tuple is
$$
(S, A, t, s, e)
$$
where

*

*$S$ is finite number of states

*$A$ is set of symbols known as the alphabet, also finite in number

*The function $t$ that operates the transition between states for each
symbol

*An initial start state $s$ where the first input is given or
processed

*A final state or states, $e$, known as accepting states.

