User talk:Mirai Nikki

we have a convention of welcoming new users via talk page, but considering that we've already talked on irc ill forgo that formality.

So, about free variables. It turns out that in definitions like \(f(x) = x^2\), \(x\) is actually a dummy variable. The reason for this is that \(f\) is a "machine" that takes real numbers and outputs other real numbers, and we only created \(x\) for the express purpose of defining \(f\). Similarly, if we say that C(x, y) = "x and y are the same color," then x and y are actually dummy variables.

Returning to relations, there are two major ways to notate them. We can use a function-like notation like \(C(x,y)\) or \(D(x,y)\), but we can also write \(x\,D\,y\) or \(x\,R\,y\). Here \(D\) and \(R\) are treated as operators similar to \(<\) or \(\leq\) or whatever.

Okay so let's look at a specific type of relation called a total order. A total order satisfies the following properties:


 * Antisymmetry: For all \(a\) and \(b\), if \(a\,R\,b\) and \(b\,R\,a\) both hold, then \(a = b\).
 * Transitivity: For all \(a\), \(b\) and \(c\), if \(a\,R\,b\) and \(b\,R\,c\) both hold, then \(a\,R\,c\).

This may seem cryptic, but look what happens when we let \(R\) be \(\leq\):


 * Antisymmetry: For all \(a\) and \(b\), if \(a \leq b\) and \(b \leq a\) both hold, then \(a = b\).
 * Transitivity: For all \(a\), \(b\) and \(c\), if \(a \leq b\) and \(b \leq c\) both hold, then \(a \leq c\).

Both of these are true, for real numbers or rational numbers or integers or natural numbers. Indeed, every total order intuitively works sort of like \(\leq\).

Let's look at a total order that doesn't use numbers. consider the set of letters {e,i,o,p,q,r,t,u,w,y} and let \(x \preceq y\) be "\(x\) is to the left of (or is the same letter as) \(y\) on the top row of the QWERTY keyboard." It is not hard to show that this is a total order. (\(\preceq\) is a curved less-than-or-equal-to sign, used to denote a relation that's similar in behavior to \(\leq\).) it's vel time 22:25, October 12, 2014 (UTC)