Lecture 002

Set Notation

Set(unofficial): a set is unordered collection of objects.

Roster Method: list of elements in the set (informal, typically written when sets are small and finite)

Variable Proposition: a statement involving unquantified variables which takes a truth value for appropriate values of the variables.

Set Builder Notation

Important Number Sets:

Integer Divisibility: for a, b \in \mathbb{Z}, we say a divides(be inversely divisible by) b denoted a|b, if and only if there exist an integer c\in\mathbb{Z} such that b=ac.

Even and Odd: n\in\mathbb{Z} is even if 2|n, and is odd if 2|(n-1).

Rational Numbers: \mathbb{Q}

Real Numbers: \mathbb{R}

:warning: integer exponentiation is not within group because there are negative exponents

Empty Set: \{\} or \emptyset

For n\in\mathbb{Z}, [n] means "a set consisting of the first n natural number"

Well-Defineness of Sets

Russell's Paradox

S = \{X|X\notin X\}, then "Is S\in S"

:question: (variable preposition \{\}\in\{\{\}\} returns Yes, and \{\}\in\{\} returns No, so {} is not the same as {{}}) :warning: \in just mean the word in in python, and \subseteq means contains in python.


Subset: A is subset of B (A\subseteq B) if and only if every element of A is also an element of B. (not subset is \nsubseteq)

Equal: A\subseteq B \text{ and } B\subseteq A

Proper Subset(\subset or \subsetneq): A\subseteq B and A\neq B

\emptyset \subset [6] \subset \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

Power set: Let A be a set, \mathcal{P}(A) is the set of all subset of A. (including A itself)

:= is assignment symbol

\mathcal{P} (A\in\{X|[whatever]\} = \{A|[whatever]\})

:warning: natural numbers include 0, but [1] starts with 1.

TODO: How can we denote a pair of numbers as coordinates in set notation? (see piazza)

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