Why are finite automata defined as ordered $n-tuples$ and not as sets? First off, I should mention that I am a new contributor on this site and as such, please bear with me if this has been asked and answered before. I searched to the best of my ability to find similar questions and didn't find any.

The Question

Consider the definition of a deterministic finite automaton given below:


  A deterministic finite automaton $M$ is an ordered $5$-tuple $
 (Q,\Sigma, \delta, q_0, F)$ where:
  
  
  $Q$ is the set of input states;
  $\Sigma$ is a finite set of input symbols;
  $\delta\colon Q \times \Sigma \to Q$ is the transition function;
  $q_0$ is the initial state
  $F$ is the set of final states
  


My main question is, would it be inaccurate to define $M$ as the set $ \{Q,\Sigma, \delta, q_0, F\}$ of five elements? 
I arrived at this question when wondering if a DFA must necessarily be ordered since n-tuples are, by definition, ordered. In other words, the tuple $(Q,\Sigma, \delta, q_0, F)$ is, by definition, different from $(Q,\Sigma, \delta, F,q_0)$. But what if we define its elements the same way as given in the definition above? Will it be a DFA? If it will, then can a DFA be instead defined as a set?
A DFA fundamentally has five components, and those components aren't interchangeable. You can't freely swap the set of final states and the set of input symbols, for example, without just making a mess of everything.

The standard representation just says "those five things make a DFA". In other words, a DFA is just those five things in sequence, nothing more or less. The order matters, not because the input symbols have to come before final states, but because you need to know which is which and not get that wrong. Without an order, you wouldn't know that.

But, if you wanted to represent a DFA as some sort of dictionary instead, like $\{``Q":\{\ldots\}, ``\Sigma":\{\ldots\},\ldots\}$, that would be perfectly fine. Or a list of pairs, or really any way you like that makes it clear what's what.