Math Zero

I have always hated mathematics (thanks to my high school teacher). This is my attempt to re-learn math from zero.

Peano Axioms

$$ \begin{align} & && 0 \in \mathbb{N} \\ & && \forall n \in \mathbb{N}, S(n) \in \mathbb{N} \\ & && \forall n \in \mathbb{N},S(n) \neq 0 \\ & && \forall m, n \in \mathbb{N}, (S(m) = S(n) \rightarrow m = n) \\ & && \forall A \subseteq \mathbb{N},\; \Bigl[\bigl(0 \in A \;\land\; \forall n \in A,\; S(n) \in A\bigr) \rightarrow A = \mathbb{N}\Bigr] \end{align} $$

The axioms of Zermelo set theory

  1. Axiom of Extensionality

$$ \forall x\, \forall y\; \Bigl[ \forall z\; (z \in x \leftrightarrow z \in y) \rightarrow x = y \Bigr] $$

  1. Axiom of elementary sets
  • Axiom of Empty set

$$ \exists A\; \forall x\; (x \notin A) $$

  • Axiom of pairing

$$ \forall A\, \forall B\; \exists C\; \forall D\; \bigl( D \in C \leftrightarrow (D = A \lor D = B) \bigr) $$

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Sets, Numbers and Quantifiers

  • Empty set \(\emptyset\)

$$ \emptyset = \{ \} \quad $$

The set with no elements

  • Such that \(\mid\)

$$ A = \{ x \in \mathbb{N} \mid x \text{ is even} \} $$

A is the set of natural numbers such that \(x\) is even.

  • Universal quantifier \(\forall\)

$$ \forall x \in \mathbb{N}, \; x + 0 = x $$

For all natural numbers \(x\), \(x + 0\) equals \(x\).

  • Existential quantifier \(\exists\)

$$ \exists x \in \mathbb{N}, \; x^2 = 16 $$

There exists a natural number \(x\) such that \(x^2 = 16\).

  • Combining both

$$ \forall x \in \mathbb{N}, \; \exists y \in \mathbb{N}, \; x < y $$

For every natural number \(x\), there exists a natural number \(y\) such that \(x < y\).

  • Union \(\cup\)

$$ A \cup B = \{ x \mid x \in A \text{ or } x \in B \} $$

  • Intersection \(\cap\)

$$ A \cap B = \{ x \mid x \in A \text{ and } x \in B \} $$

  • Subset \(\subseteq\)

$$ A \subseteq B \iff \forall x \, (x \in A \Rightarrow x \in B) $$

  • Not a subset \(\not\subseteq\)

$$ A \not\subseteq B \iff \exists x \, (x \in A \text{ and } x \notin B) $$

  • Element of a set \(\in\)

$$ x \in A \iff x \text{ is an element of } A $$

  • Not an element of a set \(\notin\)

$$ x \notin A \iff x \text{ is not an element of } A $$

  • Natural numbers \(\mathbb{N}\)

$$ \mathbb{N} = \{ 1, 2, 3, \dots \} $$

Some definitions include \(0\).

  • Integers \(\mathbb{Z}\)

$$ \mathbb{Z} = \{ \dots, -2, -1, 0, 1, 2, \dots \} $$

  • Rational numbers \(\mathbb{Q}\)

$$ \mathbb{Q} = \left\{ \frac{p}{q} \;\middle|\; p, q \in \mathbb{Z},\; q \neq 0 \right\} $$

  • Real numbers \(\mathbb{R}\)

$$ \mathbb{R} = \mathbb{Q} \cup \{\text{irrational numbers}\} $$

  • Prime numbers \(\mathbb{P}\)

$$ \mathbb{P} = \{ 2, 3, 5, 7, 11, \dots \} $$

  • Complex numbers \(\mathbb{C}\)

$$ \mathbb{C} = \left\{ a + bi \;\middle|\; a, b \in \mathbb{R},\; i^2 = -1 \right\} $$

  • Purely imaginary numbers

$$ i\mathbb{R} = \{ bi \mid b \in \mathbb{R} \} $$

These are complex numbers whose real part is zero.

  • Quaternions \(\mathbb{H}\)

$$ \mathbb{H} = \left\{ a + bi + cj + dk \;\middle|\; a, b, c, d \in \mathbb{R} \right\} $$

Irrational Numbers

An irrational number is a real number that cannot be written as a ratio of two integers.

\[ x \text{ is irrational } \iff x \notin \mathbb{Q} \]

This means there are no integers \(p, q \in \mathbb{Z}\) with \(q \neq 0\) such that

\[ x = \frac{p}{q} \]