Set theory Equations and definitions

Membership and containment

• $x\in A$: $x$ is an element of $A$.
• $xotinA$: $x$ is not an element of $A$.
• $A\subseteq B⟺\forall x(x\in A\Rightarrow x\in B)$.
• $A\subset B$: often means proper subset, i.e. $A\subseteq B$ and $AeqB$.

Operations

$A\cup B=\{x:x\in A\text{ or }x\in B\}$ $A\cap B=\{x:x\in A\text{ and }x\in B\}$

otin B\}$$ $$A\times B=\{(a,b):a\in A,\ b\in B\}$$ ## Relations and functions A [[Relation]] from $A$ to $B$ is a subset $R\subseteq A\times B$. A [[Function]] is a special relation where each $a\in A$ appears with exactly one $b\in B$. ## Power set and cardinality - $\mathcal P(A)=\{S:S\subseteq A\}$ is the set of all subsets of $A$. - $|A|$ is the cardinality, or size, of $A$. - For finite $A$, $|\mathcal P(A)|=2^{|A|}$. These definitions feed directly into [[Domain]], [[Codomain and range]], probability events, and state spaces in physics. ## Set-builder notation A set can be defined by a property:

{x\in\mathbb{R}: x^2<1}=(-1,1).

Read this as: "the set of real numbers $x$ such that $x^2<1$." ## Cartesian product The Cartesian product is

A imes B={(a,b):a\in A,;b\in B}.

$$RelationsandfunctionsarebuiltfromCartesianproducts,sothislinksdirectlyto[[Relation]],[[Mapping]],and[[Function]].$$