Lecture - Analysis MT22, I

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?

\[a < b \iff b - a \in \mathbb{P}\]