21127 Lecture Notes - Lecture 8: Mathematically Correct, Natural Number, Xsan
16 Dec 2015
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Document Summary
Let"s actually write up some rigorous, mathematically correct, and well-written proofs. All of the proposi- tions contained here are useful facts that we can cite later on, and we would expect you to be able to prove claims like these. Furthermore, all of these proofs are of the type of quality and rigor that we will expect from you on homework. If a, b are sets and p(a) p(b), then a b. To prove a b, we must show that whenever we have an x a, it is also true that x b. First, consider the case where a = . Then, obviously a b, since b, no matter what set b is. Next, consider the case where a (cid:54)= . Then, we can consider an arbitrary and xed x a. We know, then, that {x} a, and thus {x} p(a). By our assumption that p(a) p(b), we may deduce that {x} p(b).
