# Mathematical Notation…

$x$ and $y$ are two different, positive and real numbers. How can you denote this using mathematical symbols for positive and real numbers.

• $x,y \in \mathbb R^+, x \ne y$. – Mauro ALLEGRANZA Oct 8 '16 at 17:22
• Of course the best way is: "$x$ and $y$ are two different positive real numbers". – GEdgar Oct 8 '16 at 19:02

First you need to settle on a symbol for real numbers. I personally prefer $\mathbb R$, but you might find $\textbf R$ occasionally. Having made that choice, it's a simple matter to append a superscript plus sign to that symbol.

Believe it or not, some people consider 0 to be a positive number, then what I consider positive numbers they consider "strictly positive" numbers. It wouldn't hurt to be clear where you stand on this.

Then to say that $x$ and $y$ are distinct numbers drawn from this set, you could write $0 < x < y$ (or $0 \leq x < y$ for the positive zero weirdos). The problem with this is that it precludes the possibility that $y < x$. Better then to write $x \neq y$ if you don't require one to be larger than the other, merely distinct.

Therefore,

$x, y \in \mathbb R^+$, $x \neq y$.

is the best way to go about it, in my opinion. Make sure to state in words what $\mathbb R^+$ stands for at some point.

1. $x\neq y\in \mathbb{R}_+$
2. $x,y\in\mathbb{R}_+, x\neq y$
3. Since $\mathbb{R}$ is totally ordered, we have trichotomy so we can say $0 < x < y \in \Bbb{R}$

There are many more ways you can do this.

• The "+" goes on top, like so: $\mathbb{R}^{+}$. – barak manos Oct 8 '16 at 17:34
• BTW, the 3rd bullet is wrong IMO. Should be $(0<x<y)\vee(0<y<x)$. – barak manos Oct 8 '16 at 17:35
• @barak manos I will agree it is more common to see "+" on top, but it is not required. It is a preference of the author and in so doing is commonly defined at the beginning of a paper that uses it. The third bullet isn't wrong. If you are declaring to choose two distinct numbers you can label them what you will. BUT if you are given two numbers you can use what you said or a WLOG argument. – G. Snapsmath Oct 8 '16 at 17:46
• Aside from the $\mathbb R^+, \mathbb R_+$ quibble, I like the second one best. – Robert Soupe Oct 8 '16 at 18:09
• 1 seems wrong since $x$ could be negative. – Theorem Oct 8 '16 at 18:34