21127 Lecture Notes - Lecture 7: Die Tageszeitung
16 Dec 2015
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It helps to think about these standard objects of mathematics. Practicing abstraction, and abstract thinking, is very helpful! For every n n, we let [n] denote the set. This is convenient notation to clean up summation notation: [n] = {x n | 1 x n} k = n(n + 1) Suppose m, n n and m n. observe that [m] [n]. We need to know [m] [n] and [m] [n]. Oh look, that"s the same as m n and m n, neat! In general, we can prove the following claim: Suppose that a and b are sets and a b. We want to show that p(a) p(b). To do that, we need to satisfy the formal de nition of subset . That is, we need to show that every element x p(a) also satis es x p(b). To prove this, let"s take an arbitrary and xed element x p(a). We need to explain why x p(b), as well.
