147Pm SAGA

1 ‌2017-01-08

2 2017-01-09T19:46:56; M590

2.1 Notation practice

  1. For every \(x\) there is a \(y\) such that \(x < y\) becomes

    \[ \forall x \, \exists y \: | \: x < y \]

    For every \(\epsilon\) there is a \(\delta\) such that for every \(y\)

    \[ if \: \left|x - y\right| \: < \delta \quad then \quad \left| f(x) - f(y) \right| \: < \: \epsilon \]

    becomes

    \[ \forall \epsilon \, \exists \delta \, \forall y \, (\left|x - y \right| \: < \: \delta \rightarrow \left|f(x) - f(y)\right| \: < \: \epsilon) \]

2.2 Axiomatic set theory

  1. Equality

    Two sets are equal if and only if they have the same elements. Moreover, equal elements belong to the same sets.

2.3 Mathematical Induction

  1. The Well-Ordering Principle

    Every nonempty set \(S\) of nonnegative integers contains a least element; that is, there is some integer \(a\) in \(S\) such that \(a \leq b\) for all \(b\) 's belonging to \(S\).

  2. Proof: Let \(A\) be a non-empty subset of \(\mathbb{N}\). We wish to show that \(A\) has a least element, that is, that there is an element \(a \in A\) such that \(a \, \le \, n\) for all \(n \in A\). Do strong induction on the following predicate:

    \(P(n)\) : If \(n \, \in \, A\), then \(A\) has a least element.

    Basic step: \(P(0)\) is clearly true, since \(0 \le n\) for all \(n \in \mathbb{N}\).

    Strong Inductive Step: We want to show that \([P(0) \land P(1) \land \dots \land P(n)] \rightarrow P(n + 1)\). Then suppose \(P(0), P(1), \dots, P(n)\) are all true and that \(n + 1 \in A\). Consider two cases:

  3. \(\mathbb{Z}\)

Date: 2015-05-09 Sat 07:53

Author: 147Pm

Created: 2017-01-10 Tue 08:09

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