01 - Set Theory - lec. 1

ucla | MATH 170E | 2023-01-09T16:16

Table of Contents

Definitions
Big Ideas
- Probability theory (statistics)
  - Law of Large Numbers
  - Central Limit Theorem
- Set theory (review)
Resources

Definitions

probability space
- model for predicting randomness
random variable
- the variable of question of measurement
distribution
- the concentration of randomness

Big Ideas

Probability theory (statistics)

The practice of quantifying randomness

Set theory is important in creating concrete explanations of randomness

Law of Large Numbers

the expected distribution over the course of many tries, the long-term,

Central Limit Theorem

the distribution of randomness in the long-term given some specific constraints will result in a bell-shaped curve (Gaussian distribution)

Set theory (review)

Set difference

denoted $B ∖ A \equiv B - A$

Universal set and Complement

denoted $Ω$
complement of $A$ denoted $A’=\Omega A$

Formulaic unions

given sets $A_1,…,A_k k\in\mathbb{N}$
union $\cup_{j = 1}^{k} A_{j} = A_{1} \cup \dots \cup A_{k}$

Formulaic intersections

$\cap_{j = 1}^{k} A_{j} = A_{1} \cap \dots \cap A_{k}$

DeMorgan’s Law

if $A_j\sube\Omega j=1,…,k,…,\infin$ then
$(\cup_{j}^{k} A_{j})^{'} = \cap_{j}^{k} A_{j}^{'}$
$(\cap_{j}^{k} A_{j})^{'} = \cup_{j}^{k} A_{j}^{'}$

Resources

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**SUMMARY
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