What is [math] \circ [/math] when...
1. [math] a \circ b \ne b \circ a [/math]
2. [math] a \circ 0 \ne a [/math]
3. [math] 0 \circ 0 \ne 0 [/math]
4. [math] \left( {a \circ b} \right)\left( {c \circ d} \right) = ac \circ bd [/math]
5. [math] {\left( {a \circ b} \right)^{ - 1}} = {a^{ - 1}} \circ {b^{ - 1}} [/math]
6. [math] \left| {a \circ b} \right| = ab [/math]
>>7709158
if we ignore 6, that should be /
>>7709182
Wrong
>>7709158
If ab<0, then 6 is impossible.
>>7709272
You are assuming things you shouldn't be.
>>7709158
This puzzle is a bit unclear.
Which of the operations and constants used in these axioms are unspecified, and which ones are well-known operations? The [math]\textdegree[/math] is obviously unspecified, but what about [math]0[/math], [math]{}^{-1}[/math], [math]|\_|[/math], and the implicit-product operations? Do they represent the regular operations (zero, multiplicative inverse, some form of magnitude, multiplication), or are they free? Maybe they are neither this well-defined nor fully free, say with the apparent-inverse still being the inverse of the apparent-multiplication but the latter not necessarily meaning real multiplication?
>>7709158
Ok.
[math] \circ [/math] is [math] \oplus [/math]
where [math] \oplus [/math] is the matrix direct sum.
So [math]a \circ b = a \oplus b = \left( {\begin{array}{*{20}{c}}
a&0 \\
0&b
\end{array}} \right)[/math]
and [math] \left| {a \circ b} \right| = \left| {a \oplus b} \right| = \left| {\begin{array}{*{20}{c}}
a&0 \\
0&b
\end{array}} \right| = ab [/math]
>>7709158
By 2 and 3 you mean there is no identity element? Or is it the number 0?
>>7709293
What about 3?
>>7709303
[math] 0 \oplus 0 = \left( {\begin{array}{*{20}{c}}
0&0 \\
0&0
\end{array}} \right) [/math]
>>7709275
Well you have absolute value in the left hand side of 6. So I assume a \circ b is a complex number. On the right you have "ab" which must then be a complex number, so I assumed a and b were complex and this is standard multiplication.
Basically what this guy said...
>>7709290
This is very clever!
>>7709293
Except it again ambiguous what 3 means. If the right hand side of 3 means the zero matrix, then 3 is true.
>>7709158
since [math] 0 \circ 0 \ne 0 [/math] we can have [math] \left(0 \circ 0\right)^{-1} [/math] but then by 5, [math] \left(0 \circ 0\right)^{-1} = 0^{-1} \circ 0^{-1} [/math]
>>7710149
I think there are liberties being taken with [math] \ne [/math] here, where the left and right hand sides aren't even the same type of object. Like writing [math]\pi\ne C^\infty(\mathbb{R})[/math], it's true but goofy.
>>7710149
Which is true, both sides are undefined.
>>7710576
Left side isn't. Read it again.
>>7710149
so this only shows that aob is not the same type of object as a or b
