Yeshivah of Flatbush After-School Math Enrichment 2009 Jerry B. Altzman, Ph.D.

Why are we here? You've all shown an aptitude and and interest for math. There's after school writing, why not math? Because, we're not somewhere else. Duh!

What are we not doing here? We're not going to be doing lots of repetitive arithmetic. We're not going to be covering the regular math syllabus. Lots of homework, rote memorization. Huge amounts of lecturing. Occasional rants and pontifications by Jerry excepted.

Well then, what are we going be doing? Beginning the process of thinking like mathematicians: deal with abstraction and solidifying Covering stuff you've probably not seen before (or if you have seen it, not in the way you've seen it.) Like what? Set theory Logic—formal and informal Number theory Topology Games and game theory. Algorithms and how computers actually compute Codes and encryption Silly math jokes.

What Does It Mean: “Think Like a Mathematician”? “A mathematician is a machine for turning coffee into theorems”—Paul Erdős, quoting Rényi. Look for new structures. Can this specific instance of something new be generalized to a whole new class of things? Legitimately? Always? Can this general idea be narrowed down to some specific case, or is our specific case special? Can you prove it? (That's very important!) Is your proof “ugly” or “elegant”? (This is also important)

Questions We Ask As Mathematicians Can you prove that statement? If it's true over here, is it true over there? Is this the best way of doing it? Can you make a computer do it? Is it exact? If it's not exact, where does it go wrong? (If some of this stuff sounds like learning גמרא, well, it is a bit like it.)

By The End Of The Course Able to identify logic “embedded” in real life Able to identify the misuse of logic Look at the world slightly differently Be able to read some math papers for real. Understand Jerry's silly math jokes.

Books I Use (and you should read, too) The I Hate Mathematics Book Just about anything by Smullyan—logic puzzles galore! Innumeracy Others as I introduce them.

Set Theory Sets: big bags o' things. { }

Set Theory Two ways to describe sets List every element in them. Write down a rule describing the set  { all the boys in math enrichment }  { all the girls in math enrichment }

Set Theory So, what can we say about sets? What can we do with them? Count them up Mash them together Figure out what makes them different. Determine if they are they equal Not a whole lot else...

Set Theory And yet, sets are pretty powerful things!  All of arithmetic can be derived from just those three things. Careful not to over-generalize.

Set Theory Go back to our original sets. Set #1: S1={ } Set #2: S2={ all the boys in the class } Set #3: S3={ all the girls in the class } Set #4: S4={ all the students in the class } What are the sizes of each? Call this the “cardinality”

Set Theory What are all the common elements, together, between S1 and S2? Between S2 and S3? S3 and S4? Call this the “union.” Write it like this: 

Set Theory What about only the elements that sets have in common—call this the intersection? What is the intersection of S4 and S3? S1 and S2? S3 and S2? What are their sizes?  S4  S3? S1  S2? S3  S2 Sometimes it's nice to have a way to write “a set with no elements at all”: 

Set Theory Other handy things to be able to describe  Membership   Subsets / contains   Equal and not equal =   Not a member / not a subset   Sets can contain other sets!  { 1, 2, 3 } is not the same as { 1, {2, 3} }  Cardinality of the first is 3, of the second is 2

Set Theory WHOA HORSIE! If I haven't confused you yet, I'm just not trying hard enough!

Set Theory More things we'd like to say about sets:  Can you order the sets?  Clearly with our sets of people, no. But this is math, so we'll talk about numbers a lot.  In fact, once you introduce order, and one more thing, you've got pretty much all of math wrapped up. Time to go home!

Set Theory Cardinality seems so … boring. I mean, we all know how to count, right? What about some other, seemingly simple sets like:  { all the positive numbers 1,2,3,4,... }  { all the decimal or fractions you can write between 0 and 1 like ½, ¾, , }  Wait a second...we can't count up all those!

Set Theory So … let's invent something new. Typical trick: find a new thing, name it!  card { all the counting numbers 1,2,3,4, …} is … 00  Not quite the same thing as , but close enough for the time being.

Set Theory From all of this, arithmetic!  (With one or two other little things from a guy named Peano.) But it won't look quite right. But don't worry, we'll rederive arithmetic later and it will look just as wrong!