MIT6_042JF10_lec13

MIT6_042JF10_lec13 - "mcs-ftl 2010/9/8 0:40 page 379#385 13...

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13 “mcs-ftl” — 2010/9/8 — 0:40 — page 379 — #385 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite set does have an infinite number of elements, but it turns out that not all infinite sets have the same size—some are bigger than others! And, understanding infinity is not as easy as you might think. Some of the toughest questions in mathematics involve infinite sets. Why should you care? Indeed, isn’t computer science only about finite sets? Not exactly. For example, we deal with the set of natural numbers N all the time and it is an infinite set. In fact, that is why we have induction: to reason about predicates over N . Infinite sets are also important in Part IV of the text when we talk about random variables over potentially infinite sample spaces. So sit back and open your mind for a few moments while we take a very brief look at infinity . 13.1 Injections, Surjections, and Bijections We know from Theorem 7.2.1 that if there is an injection or surjection between two finite sets, then we can say something about the relative sizes of the two sets. The same is true for infinite sets. In fact, relations are the primary tool for determining the relative size of infinite sets. Definition 13.1.1. Given any two sets A and B , we say that A surj B iff there is a surjection from A to B , A inj B iff there is an injection from A to B , A bij B iff there is a bijection between A and B , and A strict B iff there is a surjection from A to B but there is no bijection from B to A . Restating Theorem 7.2.1 with this new terminology, we have: Theorem 13.1.2. For any pair of finite sets A and B , j A j ± j B j iff A surj B; j A j ² j B j iff A inj B; j A j D j B j iff A bij B; j A j > j B j iff A strict B:
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380 “mcs-ftl” — 2010/9/8 — 0:40 — page 380 — #386 Chapter 13 Infinite Sets Theorem 13.1.2 suggests a way to generalize size comparisons to infinite sets; namely, we can think of the relation surj as an “at least as big” relation between sets, even if they are infinite. Similarly, the relation bij can be regarded as a “same size” relation between (possibly infinite) sets, and strict can be thought of as a “strictly bigger” relation between sets. Note that we haven’t, and won’t, define what the size of an infinite set is. The definition of infinite “sizes” is cumbersome and technical, and we can get by just fine without it. All we need are the “as big as” and “same size” relations, surj and bij, between sets. But there’s something else to watch out for. We’ve referred to surj as an “as big as” relation and bij as a “same size” relation on sets. Most of the “as big as” and “same size” properties of surj and bij on finite sets do carry over to infinite sets, but some important ones don’t —as we’re about to show. So you have to be careful: don’t assume that surj has any particular “as big as” property on infinite sets until it’s been proved.
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