Trying to explain college math :\

[ToffeeC]

Ok so I'm tutoring for a class this term which is supposed to be an intro to mathematical proofs. One kid asked me how to understand sentences like "Let x and y be real numbers then x = y or x =/= y.". I'm over simplifying, between - it's the use of anonymous variables that seems to bug them. I told them to read it as "for any real number values x and y take, the following holds". Not sure if it was the proper way to explain it, but they came back at me with the more complicated question of universal generalization, e.g. proving something for an arbitrary x and concluding it holds true for all such x. I'm having a really hard time explaining those things. Although they are implicitly clear to me, I can't really make them clear for others. Can anyone give me an explanation for those two things that beginners would understand? Also, can someone give me tip about explaining this kind of stuff?

 

[Mark44]

For your first example, all the statement is saying is that if x and y are any two numbers, then the two numbers are equal are they aren't. Try using a different context, such as the names of people. Given any two people, then either they both have the same first name or they don't.

Here you are exhausting all the possibilities. For a slightly more complicated example, there's the Archimedean trichotomy. Given any two real numbers x and y, then one and only one of the following can be true:
1) x < y
2) x = y
3) x > y

A few sketches of the number line might help to get this across if working with the symbols doesn't bring immediate understanding.

If a student doesn't get the statement as expressed in symbols, try presenting the relationship in some different way: analogy, graphing, whatever works.

 

[Tac-Tics]

One kid asked me how to understand sentences like "Let x and y be real numbers then x = y or x =/= y.".

The key difficulty here looks like the use of a universal quantifier ("for all real x and y"). Quantifiers can be pretty tricky to understand, because there is no standard algebra for them. They are intuitive most of the time (but not always).

There's a good introduction to the use of logical quantifiers in Hofstadter's book Godel, Escher, Bach. You can preview the relevant pages on Amazon (it's in the Typographical Number Theory chapter).

Basically, "let x and y be real numbers, x = y or x /= y" is a shorthand for an infinite series of statements:

"0 = 0 or 0 /= 0"
"1 = 0 or 1 /= 0"
"2 = 0 or 2 /= 0"
...
"0 = 1 or 0 /= 1"
"0 = 2 or 0 /= 2"
"0 = 3 or 0 /= 3"
...
"1 = 0 or 1 /= 0"
"1 = 1 or 1 /= 1"
"1 = 2 or 1 /= 2"
...

(Of course, with this infinite list of statements spans all pairs of real numbers and cannot properly be enumerated, even in a suggestive way).

It may help the student to understand that they are allowed by the rule of universal instantiation to substitute any valid value they want for x and or y. Doing so creates a more "concrete" statement that is no less true.

For constructing proofs (as opposed to comprehending existing proofs), there are a few ways to visualize this. You can translate "let x be a real number" as "your friend is going to bring a real number to the party, but you have no clue what number he's going to bring... and he's kind of a prankster, and he'll do his best to bring a number to mess you up".

Thus, you can't make any assumptions about the number. You can't assume it's less than two billion. You can't assume it's an integer or that it's rational. What DO you know about the number? You know it's REAL (meaning, no "i" funny business or infinities). You know you can always find a number bigger than it (x < x + 1) and a number less (x > x - 1). You know that it's absolute value and its square are always non-negative. You know that if it's any number of than zero, you can take its reciprocal. You know that it's either zero or nonzero. You know it's algebraic or transcendental. You know given a set of real numbers, it's either an element of the set or its not.

Give your student those examples and try and have him come up with his own. Knowing only that a number is real does tell you a lot about it.

What is this class? It might help to know the subject to give some good pedagogical examples.