Mathematics for AI 1

Module 1: Sets, Logic, and Relations

A set is a collection of objects called elements, where $e \in A$ indicates containment and $e \notin A$ indicates non-containment. Set operations include:

Union:

A \cup B := \{x : x \in A \lor x \in B\}

Intersection:

A \cap B := \{x : x \in A \land x \in B\}

Difference:

A \setminus B := \{x : x \in A \land x \notin B\}

Complement:

A^C := U \setminus A

Logical propositions are built using the universal quantifier $\forall$ denoting "for all", and the existential quantifier $\exists$ denoting "there exists at least one". Logical predicates correspond to set theory such that the "and" operator corresponds to intersection $\cap$ , "or" to union $\cup$ , and "not" to complement $A^C$ .

A relation $R$ between sets $M$ and $N$ is a subset of the Cartesian product:

R \subseteq M \times N

Relations possess distinct properties:

Injective: $\forall (x_1, y_1), (x_2, y_2) \in R : x_1 \neq x_2 \implies y_1 \neq y_2$ .
Surjective: $\forall y \in N \exists x \in M : (x, y) \in R$ .
Bijective: Both injective and surjective.
Functional: $\forall x \in M, y_1, y_2 \in N : (x, y_1), (x, y_2) \in R \implies y_1 = y_2$ .

Relations on a single set $M$ can form an equivalence relation if they are reflexive, symmetric, and transitive. They form a partial order if they are reflexive, antisymmetric, and transitive.

Module 2: Functions, Real, and Complex Numbers

A function $f: M \to N$ uniquely assigns every $x \in M$ to exactly one $f(x) \in N$ . A function is invertible if and only if it is bijective. The composition of two invertible functions $f$ and $g$ is also invertible, adhering to the property:

(g \circ f)^{-1} = f^{-1} \circ g^{-1}

The field of real numbers $\mathbb{R}$ obeys field axioms including commutativity, associativity, distributivity, and the existence of identity and inverse elements. A subset $A \subset \mathbb{R}$ is bounded from above if:

\exists C \in \mathbb{R} \forall a \in A : a \leq C

Complex numbers $\mathbb{C}$ extend real numbers using the imaginary unit $i$ , where $i^2 = -1$ . A complex number has the canonical form $z = x + iy$ , where $x$ is the real part $\text{Re}(z)$ and $y$ is the imaginary part $\text{Im}(z)$ . The absolute value is defined as:

|z| = \sqrt{x^2 + y^2}

They can also be written in Euler form using polar coordinates:

z = |z|e^{i\phi} = |z|(\cos\phi + i\sin\phi)

Complex division is executed using the complex conjugate $\bar{z} = x - iy$ :

\frac{z_1}{z_2} = \frac{z_1 \cdot \bar{z}_2}{z_2 \cdot \bar{z}_2} = \frac{z_1 \cdot \bar{z}_2}{|z_2|^2}

Module 3: Vectors and Linear Algebra

The set of $n$ -dimensional vectors is denoted $\mathbb{R}^n$ , with elements $\mathbf{a} = (a_1, a_2, \dots, a_n)$ . The scalar (dot) product of vectors $\mathbf{u}$ and $\mathbf{v}$ is:

\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i

The Euclidean norm measures vector length and is given by:

||\mathbf{v}||_2 = \sqrt{\sum_{i=1}^n v_i^2} = \sqrt{\mathbf{v} \cdot \mathbf{v}}

The triangle inequality applies to Euclidean norms:

||\mathbf{u} + \mathbf{v}||_2 \leq ||\mathbf{u}||_2 + ||\mathbf{v}||_2

A matrix $A \in \mathbb{R}^{n \times m}$ is a rectangular arrangement of scalars. The matrix product $C = AB$ for $A \in \mathbb{R}^{n \times k}$ and $B \in \mathbb{R}^{k \times m}$ is computed via row-column scalar products:

c_{ij} = \sum_{l=1}^k a_{il}b_{lj}

The inverse of a square matrix $A^{-1}$ satisfies $AA^{-1} = I_n$ . It can be found using the adjunct matrix and the determinant:

A^{-1} = \frac{1}{\det A} \hat{A}

Alternatively, Gauss-Jordan elimination transforms an augmented matrix $[A | I]$ into $[I | A^{-1}]$ via elementary row operations.

Module 4: Systems of Linear Equations

A linear system $Ax = b$ of $n$ equations with $n$ unknowns can be solved via Cramer's rule when $\det A \neq 0$ :

x_i = \frac{D_i}{D}

However, Cramer's rule is computationally inefficient ( $O(n!)$ ). Gaussian Elimination is preferred, transforming $Ax = b$ to an upper triangular system $Ux = \hat{b}$ using multipliers:

m_{ik} = \frac{a_{ik}^{(k)}}{a_{kk}^{(k)}}

This elimination process implicitly factorizes $A$ into a lower triangular matrix $L$ and an upper triangular matrix $U$ :

A = LU

Solving $Ax = b$ then reduces to sequential triangular substitutions: $Ly = b$ (forward substitution) and $Ux = y$ (backward substitution).

A matrix represents a linear mapping, and the Rank-Nullity theorem connects its kernel (null-space) to its rank:

\text{rank}(A) + \text{nullity}(A) = n

Module 5: Sequences and Series

A sequence $(a_n)_{n \in I}$ maps an index set to a set of values. A sequence is a Cauchy sequence if its terms become arbitrarily close:

\forall \epsilon > 0 \exists n_0 \in \mathbb{N} \forall n,m \geq n_0 : |a_n - a_m| < \epsilon

In $\mathbb{C}$ , a sequence converges if and only if it is a Cauchy sequence.

An infinite series sums a sequence, $\sum_{k=1}^\infty a_k = \lim_{n \to \infty} s_n$ , where $s_n$ is the partial sum. Absolute convergence occurs when $\sum_{k=1}^\infty |a_k| < \infty$ , which implies standard convergence.
Important series proofs and tests:

Geometric Series: $\sum_{k=0}^\infty q^k = \frac{1}{1-q}$ for $|q| < 1$ .
Harmonic Series: $\sum_{k=1}^\infty \frac{1}{k} = \infty$ (divergent).
Root Test: Convergent if $\limsup_{k \to \infty} \sqrt[k]{|a_k|} < 1$ .
Ratio Test: Convergent if $\limsup_{k \to \infty} |\frac{a_{k+1}}{a_k}| < 1$ .

Module 6: Continuous Functions and Limits

A function $f: D \to \mathbb{R}$ is continuous at $x_0 \in D$ if it satisfies the $\epsilon$ - $\delta$ criterion:

\forall \epsilon > 0 \exists \delta > 0 \forall x \in D : |x - x_0| < \delta \implies |f(x) - f(x_0)| < \epsilon

The composition of two continuous functions $g \circ f$ is continuous. The Heavyside function serves as a classic prototype of a discontinuous function.