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PROFESSOR: The arguments that we just reviewed for proving inclusion-exclusion by looking at how many times points are counted can be made perfectly rigorous. In fact, we'll do that in a later segment, but it's interesting and good practice to realize that they can also be proved just from some simple set theoretic identities using the ordinary disjoint sum rule. So how am I going to prove the inclusion-exclusion principle for two sets? The size of A union B is the size of A plus the size of B minus the size of A intersect B, and the idea is just break up A union B into disjoint sets because once they're disjoint sets, I can add up their sizes. So if we look at A union B, A union B can be expressed as the union of two disjoint sets, namely A-- the round blue circle-- and what's left-- the points in B that are not in A, so this lighter orange-colored region. So A and B union and B minus A-- so these are the points in A and these are the points that are in B...

PROFESSOR: [? Diagonal ?] arguments are elegant, and infinite sets-- some people think-- are romantic. But you could legitimately ask what is all this weird infinite stuff doing in a course that's math for computer science? And the reason is that diagonal arguments turn out to play a fundamental role in the theory of computing. And what we're going to talk about now is the application of diagonal arguments to show that there are noncomputable sets and examine a particular one. So let's look at the class of infinite binary strings. Now, we've seen that there are an uncountable number of infinite binary strings, and that's because there was a simple bijection between the infinite binary strings and the subsets of the natural numbers-- that is the power set of n. Let's look at the infinite binary strings that we might think of and call computable strings. And what I mean by a computable string is that there's simply a procedure that will tell me w...