Sets

A set is a collection of objects.

A set is a mathematical construction which must follow the following basic axioms A set is notated by the following brackets $\{\}$ with objects separated by commas. It can also be represented by a variable, most often a capital letter $S=\{1,2,3\}$.

If I say something is “in” a set, I use this symbol $\in$ See also Logical Operators ($\wedge ,\vee ,\Rightarrow ,⟺$) See also Quantifier ($\exists ,\forall$)

Set Operations

Union

Denoted by $\cup$. It is the set which is a merging of two sets. Meaning $\forall x$ $x,y\in X\cup Y⟺x\in X\vee x\in Y$

Intersection

Denoted by a $\cap$. It is the set which only consists of elements contained in two sets. Meaning $\forall x$ $x\in X\cap Y⟺x\in X\vee y\in Y$

ZFC (Zermello-Fraenkel + Choice) Axioms

#	Name	FOL	Laymans Terms
1	Axiom of extensionality	$\forall$ sets $X,Y$ and elements $x:(x\in X⟺x\in Y)⟹X=Y$	If two sets have the same exact elements then they are equal
2	Axiom of pairing	$\forall a,b\exists Z:(a,b\in Z)$	given two elements there exists a set containing those elements
3	Axiom of subsets		If you have a set and a rule, you can make a subset constructed from elements of the chosen set with that rule applied
4	Axiom of the sum set		You can construct sets that are the the unions of multiple sets