This no doubt sounds strange, but it surely comes down to the slightly odd behavior of real numbers. In an open interval like (0, 1), there is no single real number which is less than one but is greater than all other real numbers less than one. (Proof by Cantor's diagonal: write out a list of all such numbers and you can always construct another which wasn't on the list, even when the list is infinite.) In contrast, the closed interval [0, 1] has a definite maximum element, the number 1.
This seems like a good example of poor UX in mathematics; the terms are too similar and can only be learned by memorization, the notations likewise. Good luck changing it though; Wikipedia informs me that there are no less than two ISO standards codifying it.
There is a perfectly clear notation in the Wikipedia page, however: (a,b) = {xââ|a<x<b}; [a,b] = {xââ|aâĪxâĪb}
> real numbers. In an open interval like (0, 1), there is no single real number which is less than one but is greater than all other real numbers less than one.
The rational numbers are like this too.
All the rational numbers between 0 and 1 also increase arbitrarily close to 1 with no largest element. You can prove this by simple contradiction: For every rational number x < 1, there is another rational number y = (x+1)/2 with x < y < 1. So the set of rational numbers less than 1 also has no largest element.
Actually real numbers can be defined (axiomatized) in terms of the least upper bound property -- every set of real numbers that's bounded above has a least upper bound. So you could actually have a set S of rational numbers that gets arbitrarily close to something like sqrt(2) from below. S then has no rational least upper bound -- for every rational number x greater than or equal to every element of S, there is another rational number y greater than or equal to every element of S but smaller than x.
Note that a set of real numbers need not contain its upper bound -- as you noted, a bounded open interval doesn't contain its upper bound.
I don't see how this is poor UX. If we accept your statement that the terms "open" and "closed" are too similar, what do you suggest?
Also, any notation will have to be learned. Concise ones a bit more, but there is nothing one can do about that.
And that perfectly clear notation requires you to learn quite a bit by memorization, such as the meaning of those {} brackets, the |, and the symbols â and â.
Finally, you don't need Cantor's diagonalization for a proof. It is easy(1) to show that, for any real x<1, x+(1-x)/2 is real, less than one, and strictly greater than x.
Alternatively, assuming 0<x<1, write x in decimal, and increase the first digit less than nine by one to get a larger real less than one. There is such a digit because 0.9999999âĶ is not less than one.
(1) depending on how deep you want to descend into the foundations of mathematics.
This no doubt sounds strange, but it surely comes down to the slightly odd behavior of real numbers. In an open interval like (0, 1), there is no single real number which is less than one but is greater than all other real numbers less than one. (Proof by Cantor's diagonal: write out a list of all such numbers and you can always construct another which wasn't on the list, even when the list is infinite.) In contrast, the closed interval [0, 1] has a definite maximum element, the number 1.
This seems like a good example of poor UX in mathematics; the terms are too similar and can only be learned by memorization, the notations likewise. Good luck changing it though; Wikipedia informs me that there are no less than two ISO standards codifying it.
There is a perfectly clear notation in the Wikipedia page, however: (a,b) = {xââ|a<x<b}; [a,b] = {xââ|aâĪxâĪb}