# Standard notation for building records from sets

Tuples: Tuples are of the form $(v_1, \dots, v_n)$. The are indexed by natural numbers, which means that we can access its elements by $(v_1, \dots, v_n)(i)=v_i$. One way to create a set of tuples from sets $S_1, \dots, S_n$ is by this notation: $\prod_{i=1}^n S_i$

My question: Now, I want to achieve the same for records. Records are of the form $(x_1=v_1, \dots, x_n=v_n)$, where $x_1, \dots, x_n$ are variable names. We can access its elements by those variable names: $(x_1=v_1, \dots, x_n=v_n)(x_i)=v_i$.

How can I now create a set of records from some sets? In particular, if I have sets $S_1, \dots, S_n$, what is the standard notation for $\{ (x_1=v_1, \dots, x_n=v_n) \mid \forall i \in [n]: v_i \in S_i \}$?

A possible solution: I guess it would help to embed sets $S$ into tuples by something like $(x=S) := \{(x=v) \mid v \in S\}$. Then, I could use $\prod_{i=1}^n (x_i=S_i)$ to achieve what I want.

But is this the standard way? Or are there better ways?

Edit: Some context: I want to give semantics to a programming language. For this, it is convenient to model the set of states by a set of records. If I have a state $\sigma$ and want to evaluate a variable $x$ in that state, I can simply use that variable's name, as in $\sigma(x)$.

• Why not considring simply a couple of tuples ? $((x_1,…,x_n),(v_1,…,v_n))$ – Mauro ALLEGRANZA May 5 '17 at 14:04
• This sounds to me as if you are trying to find mathematical language for a programming construct (for example, the ability to specify arguments with name-value pairs rather than by position). There may not be a "right" answer. It might help if you provided more context, perhaps a use case. – Ethan Bolker May 5 '17 at 14:10
• @MauroALLEGRANZA I think that would still be quite inconvenient. In order to define what I want (in my notation $\prod_{i=1}^n (x_i = S_i)$), I would have to write something like (in your notation) $\{ (x_1, \dots, x_n) \} \times \prod_{i=1}^n S_i$ – Peter May 5 '17 at 14:23

A record can be considered a function defined on the set of variable names. So you would define the set $I =\{x_1, \dots,x_n\}$ of variable names and write $S_x$ for the set of values the variable $x$ can take. The set of records is written $\prod_{x\in I} S_x$. For a record $r$ in this set, you can use $r(x)$ or $r_x$ for the value of the entry $x$.

The notation $(x_1 = v_1,\dots,x_n=v_n)$ for a record would not be standard, but you could certainly define it. (I would use something like $(x_1\mapsto v_1,\dots,x_n\mapsto v_n)$ instead.) If your variable names are ordered (as they implicitly are by their names $x_1$,$x_2$, etc. you might also get away with the shorthand $(v_1,\dots,v_n)$, leaving the variable names implicit.

• Your answer is very helpful if the sets $S_i$ are themselves indexes by variables, as in $S_{x_i}$. But what if both the sets and the variables are indexed by integers, as in $S_1, \dots, S_n$ and $x_1, \dots, x_n$? – Peter May 7 '17 at 14:25
• I would try to change the indices of the sets $S_i$ (I don’t know how much influence you have there); the indices of the variables do not matter in this case, you can forget about them after defining the set $I$. If these indices are important to you for some reason, you can also define a helper function $h$ that maps each variable to its index, i.e. $h(x_i)=i$. Then the set of records is $\prod_{x\in I} S_{h(x)}$ or you can use $h$ to index into tuples using variables, i.e. a record is the composition $(v_1, \dots, v_n)\circ h$ and its value at $x_i$ is $(v_1, \dots, v_n)(h(x_i))=v_i$. – Eike Schulte May 7 '17 at 18:34

The notion of a "variable name" seems a lot to me like a random variable, which is a function from a set of outcomes $\Omega$ to a set of real numbers. So each variable $x_k$ would be a function $x_k:\Omega\to\mathbb R$, i.e. you'd have a rule that tells you the value of $x_k(\omega)$ for each $\omega\in\Omega$.

Given sets $S_1,...,S_n$, your desired set is expressed as

$$\{\omega\in\Omega:(x_1(\omega),...,x_n(\omega))\in S_1\times\ldots S_n\}$$

• Since I want to use this in the context of semantics for a programming language (see edit), this is not really what I want... – Peter May 5 '17 at 14:19