Unambiguous set abstraction (set-builder) notations with parameters

by Alexey   Last Updated August 10, 2018 13:20 PM

I know two common variants of set abstraction notation. An examples of the first variant is $$\{\,(x, y)\in\mathbf{R}^2\,|\,x^2 + y^2 = 1\,\},$$ and an example of the second is $$\{\,(\cos t,\sin t)\,|\,t\in\mathbf{R}\,\}.$$ I think that in common mathematical tradition, these two "set-builder" expressions unambiguously denote the same set.

The first form of the notation can always be used instead of the second, for example: $$\{\,(\cos t,\sin t)\,|\,t\in\mathbf{R}\,\} = \{\,(x, y)\in\mathbf{R}^2\,|\,\exists t\in\mathbf{R}: x =\cos t\wedge y =\sin t\,\}.$$ However, the "translated" expression is noticeably more verbose.

On the contrary, the first form can be translated into the second without much "overhead": $$\{\,(x, y)\in\mathbf{R}^2\,|\,x^2 + y^2 = 1\,\} = \{\,(x, y)\,|\,x^2 + y^2 = 1\wedge (x, y)\in\mathbf{R}^2\,\}.$$

In practice, the second form of "set-builder" notation is quite convenient, but it appears to be ambiguous, unless some artificial syntactic conventions are made. For example, what set $S$ is defined in the following sentence?

Let $p = 1$ and $S = \{\,(p + q, p - q)\in\mathbf{R}^2\,|\,pq = 1\wedge p\in\mathbf{R}\wedge q\in\mathbf{R}\wedge 1 < 2\,\}$.

I see two equally legitimate values for $S$ so defined:

1. $S =\{(2, 0)\}$ or

2. $S =\{\,(x, y)\in\mathbf{R}^2\,|\,x^2 - y^2 = 4\,\}$.

Does there exist an unambiguous form of the second variant of "set-builder" notation?

For reference, here is how some programming languages deal with a similar issue in list comprehension. More precisely, in programming languages there is no issue, because iterating over a collection and testing a predicate are two different operations in programming, while they are indistinguishable in mathematics.

[ (m, n) | m <- [0..5], 0 <= n, n <= 5, m^2 + n^2 == 25 ]


is the list of all pairs $(m, n)$, where $m$ is an integer from $0$ to $5$, $0\le n$, $n\le 5$, and $m^2 + n^2 = 25$. Note that this very sentence in English is ambiguous! The above expression in Haskell is only valid if $n$ is defined. The value of this expression is [(3,4)] if the value of $n$ is 4, but it is [] (the empty list) if the value of $n$ is 2, for example.

A similar definition in Python will be:

[ (m, n) for m in range(0, 6) if 0 <= n and n <= 5 and m**2 + n**2 == 25 ]


However, the analogous "set-builder" notation $$\{\,(m, n)\,|\,m\in\{0,\dotsc,5\}\wedge 0\le n\le 5 \wedge m^2 + n^2 = 25\,\}$$ is ambiguous.

For my personal needs, I start considering using my own notation $\{\ldots|\ldots|\ldots\}$ with bound variables listed in the middle, like so: $$A =\{\,(\cos t,\sin t)\,|\,t\,|\,t\in\mathbf{R}\,\},\\ B =\{\,(x, y)\,|\,x, y\,|\,x^2 + y^2 = 1\wedge (x, y)\in\mathbf{R}^2\,\},\\ C =\{\,(m, n)\,|\,m\,|\,m\in\{0,\dotsc,5\}\wedge 0\le n\le 5 \wedge m^2 + n^2 = 25\,\},$$ but isn't there some kind of a standard alternative?

Tags :