vertical bar for divisions not xor

2.7, 2.15: adda/vertical bar operator uses:
. the vertical bar operator has several uses
that are quite similar across most fields:
number theory:
. n|m is a truth function: does m divide n evenly?
because of that,
I wondered if it could double as the div operator;
ie, (int / int) -> Q (float or rational);
whereas (int | int) -> Z {truncate, floor, ceiling, ...}
. if you consider context, there is indeed
precidence for (n|m) as a div operator:
in number theory, (n|m) assumes
the result is being passed to a truth variable .
. this is also how (=) doubles as {equals, becomes}
in the basic.lang (eg, if a=b then a=c);
case depending on type:
# number? integer quotient;
# truth? mod = 0 .
. notice though, that the question of divisibility
is actually depending on the mod operation ...;
set theory:
{ f(x) | x`range } -- set generators
f(x) | (x in range) -- definite integrals .
-- like number theory, it concerns division:
the set is infinite until divided by
the finite range of its control variable .
. (a|b|c) means pick exactly one,
reminding me of unique existence;
however, in systems programming
there is often a need to apply both div's
and bit-wise xor's to the same ints;
eg, n xor (n div 31);
so, I'm wondering if I can find
something besides (|) for xor,
since it already fits nicely with div .

. math's existence operator(∃​) adds a bang(!)
to express a unique existence:
eg, ( for some! x: p(x,y) )
would mean
( for exactly one x: p(x,y) );
. notice how math uses separate operators
whenever there is quantification
or control var's involved;
 even though it could be more elegant
if it reused set generators:
eg, +(i.int|i = 1...n)
is a summation over 1..n;
math would prefer to present this as:
∑ (i)
i = 1 .
. likewise, math's ∃x(p(x)) can also be
expressed as
or( p(x)| x in universe );
\/ is math's operator for logical-or;
in the pursuit of minimal language,]
a good symbol for unique existence
could be \! .
. xor is used often in computingand is like unique existence
but only for the special pair-wise case:
. the composition of xor's for reducing
a list larger than a pair
is not generally equivalent to an (\!);
because, recursively, an xor's return of false
may indicate either {many, none};
while a return of true indicates uniqueness;
so then here is a counter example
(existence but non-unique):
(true xor true) xor (true xor false)
(many) xor (unique) =
(false) xor (true) = true
but true was supposed to mean unique;
whereas this was an example of many,
which should have registered as false
if testing for unique existence .
. math`s symbol for xor is a circled plus
reminding us it functions as
clock addition modulo 2 .
. if restricting the arg of (\!) to pairs
a computer's xor operation can implement
unique existence .