Benjamin Mako Hill blogged about math relation symbols in unicode.

Since he is known for his funny blog entries, I'm not sure how seriously to take his posting. ;-)

For the "neither less-than nor greater-than" symbol (≸) you should not assume that sets are always totally ordered. While this holds for the natural and real numbers, it does not hold for any ordering of the complex numbers (compatible with arithmetic): there is no total ordering there.

It is not difficult to show that in the complex numbers, i ≸ 1 (if you want -a < 0 <=> 0 < a to hold, that is. You can of course define an arbitrary ordering, but it won't play together with your arithmetic)

Another example, and probably easier for computer guys to understand is the substring relation. Let's write "foo" < "foobar" whenever "foo" is a substring of "foobar". As you can see, "foo" < "foobar" and "bar" < "foobar". But "quux" ≸ "foobar". Oh, and they are not equal, either.

Yes, you could sometimes need a "not compareable to" symbol. Maybe there is one, actually. But most often, mathematicians will just choose whichever symbol they prefer.

For the next symbol, "strictly equivalent to" (≣). I can only guess here. It depends very much on the notions of equivalence and strict equivalence you use. Don't restrict yourself to equivalence in being the same modulo some value, or to "behaving the same way".

A real-world computer example for equivalence would be the following:

Two programs are called equivalent, if they produce the same output for valid inputs. Two programs are called strictly equivalent, if they also produce the same output for invalid input. I think everyone can see that there is a difference, but ≣ < ≡. ☺