# ^{147}Pm SAGA

## 1 2017-01-08

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

### 2.1 Notation practice

*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

**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

**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 \(S\).

*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:- \(\mathbb{Z}\)