Lecture 13 21 127 Concepts of Math 09 25 2012 Suggestions and Comments on Proof Writing Always distinguish between what we want to show and what we currently know Keep track on the side of your goal and your assumptions and fill in the gaps I like to think of the what we currently know as a Bag of Tricks The bag changes over the course of a proof and is different from proof to proof but the goal of any proof is to add the desired conclusion what we want to show to our Bag of Tricks Specifically consider the following table that summarizes when we should Let a variable be arbitrary and fixed versus when we should Define choose a variable It depends on whether we know or want to show WTS a quantified statement and what quantifier that is WTS Know Let Choose Define Let Check out the Proof Strategies sheet I posted on the website for more info about this One More Good Proof The order matters in Cartesian Products Proposition Let A and B be any two sets Then A B B A A B A B Restate claim logically It s already given in symbolic logic format We ll just have to remember to let A and B be given in the first place Note It s worth thinking about how this claim is different than the following For any two non empty sets A and B we have A B B A A B This second one is weaker in the sense that it doesn t convey as much information Why is this the case Think about it and talk to me if you don t understand Proof strategy We must prove a claim so we should prove and First we ll prove We ll do this by contrapositive Supposing A B B A doesn t seem to tell us anything helpful about whether A or B or A B We ll suppose A 6 B A 6 B 6 We ll have to show A B 6 B A Second we ll prove We ll do this directly We ll suppose A B A B This gives us three cases although we can explain why two cases are very similar 1 In each case we ll show A B B A Proof Let A and B be any sets We will first prove A B B A A B A B by contrapositive That is we will show A B A B A B B A which is logically equivalent to A 6 B A 6 B 6 A B 6 B A by DeMorgan s Law Suppose A 6 B and A 6 and B 6 Since A 6 B this means either A 6 B or B 6 A We have two cases Case 1 Suppose A 6 B This means a A a B Note we are using our assumption that A 6 here as well Since B 6 we may choose an arbitrary b B Since a B we know a 6 b Consider the set A B Notice that a b A B Now consider the set B A Notice that a b B A because a B This shows that A B 6 B A Case 2 Suppose B 6 A This means b B b A Again we are using the assumption that B 6 Since A 6 we may choose an arbitrary a A Since b A we know b 6 a Consider the set A B Notice that a b A B Now consider the set B A Notice that a b B A because b A This shows that A B 6 B A Conclusion In either case we found that A B 6 B A This proves the conditional statement given above and the desired statement follows by contrapositive Next we will prove A B A B A B A B and we will do this directly Suppose A B A B We have three cases Case 1 Suppose A B Then A B A A and B A A A Thus A B B A Case 2 Suppose A Then A B and B A Note we stated this in lecture when we defined Cartesian Products Thus A B B A Case 3 Suppose B Then A B and B A for the same reason cited in the previous case Thus A B B A Conclusion In any case we found that A B B A This proves the conditional statement given above Overall we have proven the stated biconditional claim 2 Summary This proof was slightly harder because of the cases that came up We had to consider each case separately and reach the same conclusion in each case You might notice that sometimes the cases were quite similar Essentially we could swap the letters and produce the same arguments This is where the phrase without loss of generality or in shorthand WOLOG can be useful if we can guarantee that two parts of a proof will be identical up to the naming of some letters then we can invoke this phrase and save some writing We have to be very sure that the arguments will truly be identical For example we could have said in the second half of the above proof Suppose A B A B For the first case suppose WOLOG A The proof with B is identical We want to emphasize that you should be extra careful when using WOLOG you should make sure for yourself that the two arguments you would have written actually are identical and you should indicate to the reader that this is the case 3 Analyzing Proofs Squares and Roots Claim If 0 x y 2 2y x 2 then 2x 3y The claim should quantify its variables What are x and y It should really say x y R 0 x y 2 2y x 2 2x 3y We ll think later on about whether this claim is even True For now the best we could do is try playing around with some values of x and y and seeing what happens Spoof Since all squares are non negative x y 2 and 2y x 2 are non negative This is valid Not sure where it s going yet but it s valid Suppose that 0 x y 2 2y x 2 Okay it looks like we are proving the If then statement by assuming the hypothesis and looking to deduce the conclusion This is fine p p Then it follows that x y 2 2y x 2 Okay this is valid but it only works because all the terms are positive The writer really should say that when they make this step That is they should say Since all the terms are positive it follows that Thus x y 2y x No This does not follow In general always outputs the positive root So if we have a square root of a square we get z 2 z Essentially we can t cancel the square via the square root At best here we can deduce 0 x y …