Importance of the Order of Quantifiers

I remember at the beginning of CSC165, there was a stark warning to pay attention to the order of quantifiers in writing a claim. It seemed obvious at the time, but finally in A3 I still erroneously proved a claim by mixing up the claim.

Fo all e in R+, there exists d in R+, for all x, y in R+, |x-y| > d => |x+y| > e

From the second assignment, I'd been confused about how to prove a claim when there are different variables in the antecedent and consequent, but I fortunately learned that the way to connect the two was to write one of the variables in terms of the other (d, e in this case).

Remembering this, I excitedly got to work with the first version of my proof:

|x+y| <= e => |x-y| <= d # contrapositive

Assume |x+y| <= e

Let d = |x+y|

Then |x-y| <= |x+y|

-x+y <= -x-y

x-y > x+y

x /> x+2y # x, y in R+

Well that didn't quite work out as I'd expected...

I have no idea what I "proved" there. I thought this implied that the claim was false for a while (since that's what I'd "proved" here), until I was told that the "d" I'd picked here depends on x and y, and that can't happen if d come before x and y in the quantifier (before they're introduced), but that means I can use e.

Now it makes a lot more sense. The informal version of the proof can be written in one line:

|x-y| <= |x| + |y| # the addition of two positive numbers will always >= the subtraction of the same numbers

= |x + y| < e = d # letting d be e

Sometimes the simplicity of proofs really quite amaze me. When I was first exposed to proofs in MAT137, I'd always thought constructing proofs of my own would be out of my reach since the lines seem to have come out of nowhere. Now I see it just takes a bit of manipulation that comes with creativity and attentiveness to really understand each component of the claim, and the rest will follow. Additionally, the proofs that we've been introduced to in this course so far actually aren't as arbitrary as I'd initially thought. They actually all follow a set structure and method, and this realization has definitely made proofs a lot less intimidating.