Interroequals

[jack]

The ¿? post made me think of another symbol I like, =?= (Handwritten as a ? superimposed over an equals, I've never typed it before.)

If you have two expressions and want to work out which is bigger, the normal technique is to try to simplify them separately, or subtract one from the other and establish >0 or <0, isn't it? The point of =?= is that with = or > or >=, you assume the this line implies the next line[1], but with =?=, The next line implies this one, when you replace =?= with any of =, >, <, <=, >=

For instance, option A costs (x+1)²+x and option B costs 2x²-2, which is better?

(x+1)² =?= 2x-2
x²+1 =?= -2
x² =?= -3
x² > -3

I like it because:

* You can use both expressions together, and if half-way simplified versions cancel somewhat with each other you can take advantage of that.
* You can write the two expressions side by side, and don't have to introduce extra values.
* The status of the expression is clear: being compared to the other. If you subtract the two, you won't have a =?= reminding you to stop when it's clearly positive/negative
* It's easy to work out on the fly, but to make rigorous doesn't need to be rewritten, you can just change =?= to < and insert "<=" is implied by operators. It sort of working backwards, but it's easier than reversing your order to use conventional implication, because then your final line is inserted into your proof as if by magic, and too much magic makes it confusing to read.

Is there another symbol which does this no-one told me? Or another writing convention which has these advantages? I admit it's because I'm lazy and like things easy -- maths is hard enough in general without artificial difficulties :)

PS. It's an interesting thought. Little tricks like this (but commonly used, more useful ones) are an informal body of knowledge, the stuff it's hard to teach without watching someone do it -- in all subjects you need some amount of "apprenticeship" in addition to any amount of teaching.

[1] The tricky point many young students are taught only in retrospect is that the next line often but not always implies the current one. For instance, "x=-y" leads to "x²=y²" but the reverse (while true for *these* values of x and y) isn't a valid inference.
[2] Hey, "bar" works there!

Comments

[pseudomonas.livejournal.com]

Doesn't perl use <=> (the spaceship operator) for a vaguely similar purpose?

[cartesiandaemon.livejournal.com]

Ah, yes! That is good. I'm sure I read about it somewhere (having used some but not lots of perl) but didn't think of it because considering a comparison as a 2-valent function is more of a compsci thing, mathmos (as I was in the mode of at the time) thinking of relations as subsets of cartesian products of what might be on either side, rather than the indecator functions thereof :)

It probably could be co-opted though, for at least as much clarity as mine, and then just cross out the wrong end of it afterward :)

[mair_in_grenderich]

I've sometimes seen just a ? used where you use =?=

[cartesiandaemon.livejournal.com]

I've sometimes done that, but it always looks really unclear that they're supposed to be two sides of an equation; too often someone used '?' for a very unspecified variable or something.

[ewx.livejournal.com]

I've not encountered a common symbol for it. Though it wouldn't be that verbose to formalize it as "for any ordering R which satisfies a+bRa+c iff bRc and abRac iff bRc (a +ve) then..." or even just list the allowed values of R ecah time.

[cartesiandaemon.livejournal.com]

True, you can define it simply: assuming everyone grants you the the theoretical background even just as "Where =?= is > or < or [...]"

Then I guess the question is, is it worth introducing such a symbol? And is it sufficiently obvious that you can leave off the explanation :)