Set notation

Denoting a set as an object

When a set should refer to a an indivisible object, one typically denotes it by a single capital letter, e.g. $A=\{1,2,3\}$, and then $A$ can be used to stand for that set.

When referring to an arbitrary generic set, a typical notational choice is by the letter $S$. When several sets are under consideration, first few capitals, $A, B, C$, etc. are used.

Set-builder notation

Another form of extensional definition is set-builder notation, also called the set comprehension. It is a compact notation to describe a set, with the general sytax lookin like: $X=\{exp:rule\}$, with the colon, ":", (or sometimes, a pipe symbol, "|") standing for abstraction and being read as "such that".

For example: $A=\{x:x \in \mathbb{Z}, 2^x<32\}$ is read as "The set A contains all numbers of form two times x, such that x is an element of the integer set". One of the common abbreviations is to immediately specify the "type" of $x$: $A=\{x \in \mathbb{Z}: 2^x<32\}$

If the set comprehension contains more then one applicable rules, their relation can be spelled out in English or, more commonly, using logic symbols for negation (¬), conjunction (∧), disjunction (∨ or &), implication (→) and bijection (↔). Generally, predicate logic is used as a language to describe sets.

Definition by predicate

Elements of a set can be specified by a predicate i.e. in terms of a property (or properties) they possess.

Whether an object $x$ possesses a certain property $P$ is either true or false (in terms of classical logic), so it can be a subject of the propositional function $P(x)$.

A set can be specified by a predicate function; $S=\{x:P(x)\}$, means that $S$ is a set to whom each $x$, which possesses a certain property $P$, belongs; taht is, each $x$ for which $P(x)$ is true.

Axiom of Extensionality

If $X,Y$ are sets then $X = Y \iff (\forall Z. Z \to X \iff Z \to Y)$

meaning, if X and Y are sets, then, they are the same set iff they contain the same elements.

If and only if, iff: 1. if they are the same set, then they contain the same elements. 2. if they contain the same elements, then they are the same set.

Indexed sets

$$\bigcap_{i=0}^n A_i = A_0 \cap A_1 \cap \dots \cap A_n$$

$$\bigcap^{n+1}_{i=0} A_i = \bigcap_{i=0}^n A_i \cap A_{n+1}$$

$$\bigcup_{i=0}^n A_i = A_0 \cup A_1 \cup \dots \cup A_n$$