Flashcards
How many axioms are there in analysis for addition?
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?
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?
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?
\[a < b \iff b - a \in \mathbb{P}\]