# All sequences are sets?

I have learned that:

• A sequence is a function from a subset of natural numbers to some set A
• A function is a relation with certain properties
• A relation is a subset of a Cartesian product

It seems, therefore, that I could represent any sequence as a set like $$\left\{\left(n_1,a_1\right),\left(n_2,a_2\right),\dots\right\}$$,

where the $$n_i$$ form the subset of natural numbers and $$a_i$$ are what we would normally refer to as the elements of the sequence.

Is that correct? If so, it would help me think carefully about things like Cartesian products of sequences, functions from sequences to a set, etc., because I could think about things in terms of sets.

The Wikipedia article on sequences compares and contrasts them with sets, and I couldn't find a source that viewed a sequence as a kind of set, so I'm wondering whether my interpretation is wrong or just uncommon (or neither).

I have some exposure to real analysis and am learning game theory at the doctoral level.

• Yes, what you are talking about is known as a graph of a function: See en.wikipedia.org/wiki/Graph_of_a_function#Definition – user408858 Jan 22 '19 at 22:31
• Basically, if you go far enough into the theory, everything is sets. Except the things that can't be sets, and there are a few sets of those. – ConMan Jan 22 '19 at 22:51

That's completely correct, but likely to not be useful. It's also not uncommon, but ... only during the initial formalizing of sequences, after which we put it aside. Folks seldom think about things like cartesian products of sequences, because they don't correspond to "natural" operations on sequences. (An example of a natural operation on two real-number sequences might be "termwise addition"; another might be "interleaving.")

If I said "Your cellphone is entirely made up of atoms," it'd be true, but it wouldn't give you a clue about why the atoms were arranged in the way they were by the designer of the phone.

In much the same way, it's nice to find that something we regard as natural ("a sequence of stuff, one thing coming after another") can be represented by the things we've agreed to use as the basis for our mathematics (sets), but as soon as you realize that they can be represented this way, it's useful to go back to thinking about them the way you always have, rather than trying to always understand them in the most basic form.

In much the same way, everything in a computer is represented by 0s and 1s, but it'd be a nightmare to try to think about how the particular pixel on your screen that forms the upper left corner of the first digit of my phone number corresponds to a particular +3.3V on some wire at some instance. The joy of abstraction is that once you realize that using the voltages 0v and 3.3V (or some other standard), you can encode the values 0 and 1, and that you can build circuits that, say, compare these values, or add them, or subtract them, etc., you can stop thinking about electrons and think only about 0s and 1. Soon after that, you can think about 8-bit sequences of 0s and 1s (called bytes), and then think about particular ways of organizing these to represent certain not-too-large integers, or characters in a font, etc.

The one exception I've personally found is that when I'm given a challenging probability problem, I find it useful to see whether I can make it actually fit into the precise definitions I know for things like random variables (a function from a measure space to the real numbers (or small generalizations of this)).

Your interpretation is correct, but indeed as you state it, uncommon. The whole idea behind set theory is that everything can be described as a set. You can even describe the tuples $$(n_1,a_1)$$ as sets. A tuple $$(a,b)$$ is shorthand notation for the set $$\{\{a\},\{a,b\}\}$$.

What might be confusing here is that there is another way to assign a set to a sequence $$x_n$$, namely $$S=\{x_n:n\in\mathbb{N}\}$$. But this is more the set of points in the sequence, rather than the sequence itself. But still people like to call elements of $$S$$ elements of the sequence $$x_n$$, even though the elements of the sequence are technically of the form $$(n,x_n)$$.

In conclusion, even though everything can be described by a set, it is not always the best idea to do so.