How to correctly represent a collection with different kinds of objects?

Question

I'm in a bit of a confusion here:

I want to construct a collection that contains numbers and functions; something like this

$\{ \{1,2,3 \}, \{f,f_2 \} \}$

but I'm quite unsure about what's the correct notation for these kinds of sets. I vaguely remember that in my Linear Algebra class we'd represent Groups and Rings like this

$\{ \mathbb{Z}, + \}$

That is, a set and the operation defined on that set. But then I thought "that's kind of mixing apples with oranges, isn't it?". Maybe this is a dumb confusion, but I'd really appreciate if I could clear it up :)

Answer 1

When you're writing mathematics, your goal is to communicate ideas to human readers, so the only completely hard rule you have to follow is make sure your reader is not confused.

If you have a good technical use for having a set that contains both numbers and functions as elements, you can do that. However, in order not to leave your reader wondering whether there's a typo or they've misunderstood what you're doing, you should call out what you're doing in an explicit comment

Note that -- for technical convenience later -- this set contains both elements that are numbers and elements that are functions. We will treat $\mathbb R$ and $V\to V$ are disjoint such that there's no ambiguity about which elements are what.

(Beware in particular that we're treating the number $2$ as distinct from the function that multiplies its argument by $2$!)

... though if you need a warning like the second paragraph of this, you should probably rethink whether there's a less confusing way to present what you're doing.

If the reader wants to formalize what you're doing in pedantic formal set theory, they will have to do some extra work to make sure they're avoiding the clashes -- but since you have told them they need to do that, it's now on them to do it. (And most readers won't want to formalize your work in set theory, so putting that burden on them is not per se unreasonable).

Answer 2

Without going too deep into first-order logic, I think that a brief treatment of the notion of an $\textit{L-structure}$ associated to a first-order language $L$ may help you make sense of expressions of the form $<\mathbb{Z}, +>$.

When talking about Mathematical structures, it can be helpful to do so using only a restricted set of symbols, which we package together into a collection $L$ referred to as a $\textit{language}$. Roughly speaking: we expect to equip traditional sets with additional structure based on the symbols in the language which we choose to use when talking about these sets.

All languages are required to carry "logical" symbols allowing us to formulate basic logical statements, such as equality, quantification, disjunction and negation.

Beyond these logical symbols, two languages, say $L$ and $L'$, may differ in the non-logical symbols which they carry. For instance, $L$ may be a language better suited for talking about ordered sets (so that it contains a non-logical symbol allowing us to compare two elements), while $L'$ may allow us to talk about operations of addition or element inversion.

We distinguish three kinds of non-logical symbols: function symbols, relation symbols and constant symbols. Each function symbol and each relation symbol is tagged with a natural number $n$, called its $\textit{arity}$. An $n$-ary function is to be interpreted as a function on your set taking $n$-arguments, an $n$-ary relation is to be interpreted as a distinguished collection of $n$-tuples from your set, and a constant symbol is to be interpreted as a distinguished element of your underlying set.

Hence when introducing a language $L$, one typically lists all the function symbols, relation symbols and constant symbols contained in $L$. Suppose your language contains two function symbols $f$ and $g$, a relation symbol $r$ and two constant symbols $c_1$ and $c_2$. You may then write:

$$ L := \{f, g, r, c_1, c_2\}. $$

The next step is formalizing what it means to interpret the symbols of your language in a fixed set (and hence to endow that set with extra structure).

Having fixed a language $L$, an $L$-structure is defined to be a set $M$ together with an interpretation of every symbol in $L$. Namely, $M$ comes equipped with a function $f^M: M^n \to M$ for every $n$-ary function symbol $f \in L$, a relation $r^M \subset M^n$ for every $n$-ary relation symbol $r \in L$, and an identified constant $c^M \in M$ for every constant symbol in $M$. The superscripts are often dropped when the underlying set is clear from the context.

When referring to an $L$-structure $M$, you may introduce it together with all the data that it is required to carry as follows:

$<M, \{f^M\}, \{r^M\}, \{c^M\}>$.

That is, you make it explicit that we are not only talking about a set $M$, but about an $L$-structure $M$ equipped with interpretations for every symbol in $L$.

For instance, if you would like to talk about abelian groups, you may use the language of groups $L_{agp} := \{0, +\}$, which contains one constant symbol $0$ and one 2-ary (or binary) function symbol $+$.

An $L_{agp}$-structure is then a set $M$ equipped with a designated zero element $0^M$. For instance, by taking $\mathbb{Z}$ as your domain, you may consider the $L_{agrp}$-structure (dropping the superscripts):

$<\mathbb{Z}, 0, +>$

where $0 \in \mathbb{Z}$ is interpreted as the usual additive identity in $\mathbb{Z}$ and $+$ is interpreted as the usual binary operation on $\mathbb{Z}$ sending a pair $(n,m) \in \mathbb{Z}^2$ to its sum $n+m \in \mathbb{Z}$.

An $\textit{abelian group}$ is then precisely an $L_{agp}$-structure satisfying certain axioms (which you can formulate in first-order logic as a set of sentences known as a $\textit{theory}$).

To get a better sense of how these things fit together, perhaps you could try formulating an adequate language to talk about things like rings or ordered sets (hint: the latter requires the use of a binary relation).

The setup described above is used in Model Theory and Universal Algebra, but I also find it useful to keep in mind as a framework for the general purpose of studying "algebraic structures and structure-preserving maps between them".