I'm currently studying first-year calculus in university and thus far the concepts are very intuitive. On the other hand, I haven't had very much exposure to how notations are defined and/or how strictly they must be followed.
One thing which I'm curious about is interval notation. While it seems to be a powerful way to invoke sets of numbers, it is ordinarily defined according to inequalities. This leads to an interesting problem where I may define invalid intervals which make intuitive sense.
'Invalid' interval
$(a,b)$ such that $a > b$
Given the definition $(a, b)=\{x|x\in {\rm I\!R}, a<x<b\}$, the expansion of $(a, b)$ using inequalities would yield false statements such as $1<x<-2$.
But an intuitive understanding of what an interval is indicates that the proper definition would be 'reversible', i.e. having the property of $(a,b)=(b,a)$ and defined as $(a, b)=\{x|x\in {\rm I\!R}, min(\{a, b\})<x<max(\{a, b\})\}$.
It's implied that an interval is going to contain a set --usually $x$-- and would thus require the inequality definition to be true.
As such, I have a few questions in relation to this problem. Most importantly, is $(1,-1)$ considered valid?
Is there a preferred way to make it valid for a particular solution? Are there cases where an 'invalid' interval can be produced (legitimately and not contrived) by a mathematical operation and subsequently used to derive a solution that the definition above could not?
The expression $a<x<b$ is actually an abbreviation for the statement that $a<x$ and $x<b$. Thus, if we apply the definition $(a,b)=\{x\in\Bbb R:a<x<b\}$ to $(1,-1)$, we find that $(1,-1)=\varnothing$: there is no real number $x$ such that $1<x$ and $x<-1$. There are now several possible conventions, the most obvious being these:
You can disallow intervals $(a,b)$ with $a\ge b$.
You can allow intervals $(a,b)$ with $a\ge b$ and interpret them according to this definition, so that all of them are empty.
You can interpret $(a,a)$ according to the definition, making it empty, or disallow it, and interpret $(a,b)$ for $a\ne b$ as $(\min\{a,b\},\max\{a,b\})$.
In practice I have seen the first two used without comment, though I think that the first is more common. The third, which is the one that you want, is definitely non-standard, and I would not expect it to be used without being defined on the spot.