How to express inequality between variables? I have variables $a, b, c, d \in N^+$, how do I express the condition that all their values must be different? Also, what field of mathematics covers ways of expressing constraints? I couldn't even find an adequate way to tag this question.
 A: If you don't mind relabelling the variables so that they are in ascending order, you could write $a < b < c < d$. Otherwise, the phrase "pairwise distinct" is often used.
A: I think it's fine to just say that "$a$, $b$, $c$, $d$ are all different". This is precise, clear, and easy to understand. Someone might be able to concoct a clever way to say the same thing symbolically, but I suspect it will be far less clear. Doing mathematics doesn't require you to sprinkle strange symbols throughout your writing.
The confusing discussion in the comments seems to suggest that my approach is a good one :-)
A: $| \{ a , b , c , d \} | = 4$.
(But I would never use that.  I'd only say "$a , b ,c ,d \in N^+$ are distinct.")
A: There are fine examples already, I just wanted to mention, that is is possible to do it by single inequality:
$$\prod_{i\neq j}(x_i-x_j) \neq 0,$$
to be more specific
$$\big((a-b)(a-c)(a-d)(b-c)(b-d)(c-d)\big)^2 > 0.$$
Cheers!
A: You could phrase it
$$\{a,b,c,d\} = S \subseteq N^+ \ni \exists \;f:S\leftrightarrow \{1,2,3,4\}$$
I am naming the set $\{a,b,c,d\}$ as $S \subseteq N^+$, and saying that there must be a one-to-one mapping between set $S$ and a second set $\{1,2,3,4\}$ that we already know has four distinct members. 
A: $$\{a,b,c,d\}\subset\mathbb{N}^+$$
