https://quartz.jzhao.xyz Last 10 notes on 🪴 Quartz 4.0 Quartz -- quartz.jzhao.xyz https://quartz.jzhao.xyz/ACIT4330-Lectures https://quartz.jzhao.xyz/ACIT4330-Lectures Chapter 1 1.1 Sets and Numbers Lecture 1 - 1.1 Sets and Numbers (complimentary written notes: ACIT4330-2025-01-06-Lecture 1.rnote) Lecture 2 (complimentary written notes: ACIT4330-2025-01-09-Lecture ... Sat, 01 Mar 2025 12:58:10 GMT https://quartz.jzhao.xyz/ https://quartz.jzhao.xyz/ Chapter 1 1.1 Sets and Numbers Lecture 1 - 1.1 Sets and Numbers (complimentary written notes: ACIT4330-2025-01-06-Lecture 1.rnote) Lecture 2 (complimentary written notes: ACIT4330-2025-01-09-Lecture ... Sat, 01 Mar 2025 12:58:10 GMT https://quartz.jzhao.xyz/Definitions/Cauchy-Schwarz-Inequality https://quartz.jzhao.xyz/Definitions/Cauchy-Schwarz-Inequality Definition | (u|v) | \leq \| \, u \, \|_{2} \times \| \, v \, \|_{2} \| \, u + v \, \|_{2} \leq \| \, u \, \|_{2} + \| \, v \, \|_{2} Proof of Cauchy-Schwartz Insert a \equiv - \frac{\overline{(u ... Sat, 01 Mar 2025 12:50:46 GMT https://quartz.jzhao.xyz/Definitions/Least-Upper-Bound-Property https://quartz.jzhao.xyz/Definitions/Least-Upper-Bound-Property X \subset \mathbb{R} \exists c \in \mathbb{R} \; \text{such that} x \lt c, \forall x \in X sup(\langle 0, 1 \rangle) = 1 \notin \langle 0, 1 \rangle sup(\langle 0, 1 ] \,) sup(\mathbb{R}) ... Sat, 01 Mar 2025 12:50:46 GMT https://quartz.jzhao.xyz/Definitions/Statements/And https://quartz.jzhao.xyz/Definitions/Statements/And P \land Q Truth Table PQ=FFFFTFTFFTTT. Sat, 01 Mar 2025 12:50:46 GMT https://quartz.jzhao.xyz/Definitions/Statements/Implies https://quartz.jzhao.xyz/Definitions/Statements/Implies P \implies Q Truth Table PQ=FFTFTTTFFTTT. Sat, 01 Mar 2025 12:50:46 GMT https://quartz.jzhao.xyz/Definitions/Statements/Not https://quartz.jzhao.xyz/Definitions/Statements/Not ¬P The symbol (¬) is called negation Truth Table P=FTTF. Sat, 01 Mar 2025 12:50:46 GMT https://quartz.jzhao.xyz/Definitions/Subnet https://quartz.jzhao.xyz/Definitions/Subnet Definition A subnet of a net f: I \to X is a net g: J \to X and a map h : J \to I such that g = f \circ h and such that \forall i \in I \exists j \in J with h(j') \geq i \; \forall j' \geq j ... Sat, 01 Mar 2025 12:50:46 GMT https://quartz.jzhao.xyz/Definitions/Cauchy-Sequence https://quartz.jzhao.xyz/Definitions/Cauchy-Sequence Definition A Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses. Examples Cauchy Sequence \Sigma_{n=1}^\infty \frac{1}{n^2} = 1, \, \frac{1}{4}, \, \frac{1}{9}, \, \dots \lim_{ n \to \infty } \frac{1}{n^2} = 0 Which this sequence converges to 0, towards infinity Non-Cauchy Sequence \Sigma_{n=1}^{\infty}(-1)^n = -1, \, 1, \, -1, \, 1, \, \dots These never converge to a limit, hence it is not Cauchy. Sat, 01 Mar 2025 12:50:46 GMT https://quartz.jzhao.xyz/Definitions/Nets https://quartz.jzhao.xyz/Definitions/Nets \{ x_{i} \}_{i \in I} \; \text{NET} \; \underbrace{I}_{\text{VFOs}} \to X x \in \overline{X} \iff \exists \; \text{NET} \; \underbrace{x_{i}}_{\in X} \to x I = neighbourhoods of ... Sat, 01 Mar 2025 12:50:46 GMT