I dun a DIP, possibly the best DIP ever

[Joseph Rushton Wakeling]

On Friday, 24 April 2020 at 22:21:31 UTC, Walter Bright wrote:
> On 4/24/2020 1:55 PM, Walter Bright wrote:
>> How are things like this expressed in mathematics?
>
> I did a little research:
>
>     ∀xP(x)
>
> means:
>
>     for all x in P(x)

Minor note: I don't think it means for all x _in_ P(x).  Rather, 
here, we're evaluating P(x) for all values of x.

If P(x) is a propositional statement (i.e. a statement about x 
that can be true or false), this means that ∀xP(x) represents 
whether P(x) is true for all x or not.

I don't recall ever seeing this notation used other than to 
combine propositional statements in this way.

If you want to talk about all values of a function f(x) for all 
values of x in some set X, then it would be an _image_ of X under 
f:

     f[X] = { f(x) | x ∈ X }

... so if you want to talk about all elements of a given set, it 
would be as simple as:

     { x | x ∈ X }

which is the source of the "list comprehension" notations you get 
in various languages (e.g. pythonic `x for x in X`, but also of 
course `f(x) for x in X`).  See:
https://en.wikipedia.org/wiki/Set-builder_notation#Parallels_in_programming_languages