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Definition: Set

A set is a collection of objects, which are the elements of the set.

If $S$ is a set, and $x$ is an element of $S$, we write $x \in S$; if $y$ is not an element of $S$, we write $y \not\in S$.

An empty set is a set which has no elements, denoted $\emptyset$.

If a set $S$ contains a finite number of elements say $x_1, …, x_n$ , we write $S=\set{x_1, …, x_n}$.

Definition: Countability

If $S$ consists of infinitely many elements which can be enumerated in a list, such that there are as many elements as there are positive integers, we write $S=\set{x_1, x_2, …}$ and say that $S$ is countably infinite.

If the elements of $S$ cannot be enumerated in a list, $S$ is called uncountable.

Definition: A Set That Satisfies a Certain Property

We may want to consider the set $S$ of all $x$ that satisfy a certain property $P$. We write this as $S=\set{x|x\text{ satisfies }P}$

Definition: Subset and Supset

If every element of $S$ is an element of $T$(another set), we write $S \subseteq T$($S$ is a subset of $T$) or $T \supseteq S$($T$ is a supset of $S$).

Definition: Set Equal

$$S = T \iff S \subseteq T, S \supseteq T$$

Definition: Set Operations

We often introduce a universal set, $\Omega$ containing all objects of interest in a particular context.

The complement of $S$ is: $$S^c=\set{x\in\Omega \mid x\not\in S}$$

The union of $S$ and $T$ is: $$S\cup T = \set{x\in\Omega \mid x \in S \text{ or } x \in T}$$

The intersection of $S$ and $T$ is: $$S\cap T = \set{x\in\Omega \mid x \in S \text{ and } x \in T}$$

Definition: Disjoint Sets

Two sets are said to be disjoint if their intersection is empty.

Definition: Partition

A collection of sets is said to be a partition of a set $S$ if

1. the sets in the collection are disjoint
2. their union is $S$
Theorem

${({S^c})^c}=S$

$S\cap S^c=\emptyset$

$S\cup S^c = \Omega$

$S\cap(T\cup V)=(S\cap T)\cup(S\cap V)$

$S\cup(T\cap V)=(S\cup T)\cap(S\cup V)$

Theorem : De Morgan's Laws

De Morgan’s Laws, $S_1, S_2,…$ are sets

1. $(\underset{n}{\cup} S_n)^c = \underset n \cap S_n^c$
2. $(\underset{n}{\cap} S_n)^c = \underset n \cup S_n^c$