# Lecture - Analysis MT22, Lecture 1

> Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/1/ · Updated: 2022-10-17 · Tags: lecture, uni

- [Course - Analysis MT22](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/)

### Flashcards
How many axioms are there in analysis for addition?::
4

What’s the axiom relevant to addition in analysis about commutativity?::
$$
a + b = b + a
$$

What’s the axiom relevant to addition in analysis about associativity?::
$$
(a + b) + c = a + (b + c)
$$

What’s the axiom relevant to addition in analysis about the additive identity?::
$$
a + 0 = a
$$

What’s the axiom relevant to addition in analysis about additive inverses?::
$$
\exists - a : a + (-a) = 0
$$

How many axioms are there in analysis for multiplication?::
4

What’s the axiom relevant to multiplication in analysis about commutativity?::
$$
a \cdot b = b \cdot a
$$

What’s the axiom relevant to multiplication in analysis about associativity?::
$$
(a \cdot b) \cdot c = a \cdot (b \cdot c)
$$

What’s the axiom relevant to multiplication in analysis about the multiplicative identity?::
$$
a \cdot 1 = a
$$

What’s the axiom relevant to multiplication in analysis about multiplicative inverses?::
$$
a \ne 0 \implies \exists\frac{1}{a} \text{ such that } a \cdot \frac{1}{a} = 1
$$

What’s the distributive law/axiom in analysis?::
$$
a(b + c) = ab + ac
$$

What’s the zero law/axiom in analysis?::
$$
0 \ne 1
$$

How many order axioms are there in analysis?::
3

What is $\mathbb{P}$ in analysis?::
The set of all positive reals.

What’s the order axiom in analysis about addition?::
$$
a, b \in \mathbb{P} \implies a + b \in \mathbb{P}
$$

What’s the order axiom in analysis about multiplication?::
$$
a, b \in \mathbb{P} \implies ab \in \mathbb{P}
$$

What’s the order axiom in analysis about trichotomy?::
$$
\forall a \in \mathbb{R},\text{ either } a \in \mathbb{P}, a = 0, -a\in\mathbb{P}. 
$$

How is $a < b$ defined in analysis?::
$$
a < b \iff b - a \in \mathbb{P}
$$

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