Set Theory

Definition

A set is a collection of items. Order and occurrence does not matter. You can use set builder notation like:

$[x|x<10andx\in N]$
OR
$\{x\in S|P(x)\}$ → Read as “The set of all S such that P(x)”

Sets also have distinct Set Properties & Laws.

Complement of a set:

$A^c$ or $\bar{A}$.

Cartesian Product:

The set of all pairs of two sets. EG $A(1,2,3)\times B(a,b)=\{(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)\}$.

Set Equality:

$A=B⟺A\subseteq BandB\subseteq A$

Proper Subset:

A subset of a set bar the set itself. I don’t know how that’s particularly useful but there you go. Figure is: $\subset$

Power Set:

The power set of a set is a set with all the possible subsets of the original set. The size is $2^n$, where $n$ is the original size of the set.

Disjointed Sets:

Sets A and B are disjointed iff they share no elements in common. ($A\cap B=\varnothing$)
Mutually Disjoint Sets often refers to multiple sets (A, B, C) with no similarities amongst them.

Partition of Sets:

A set of subsets of A, where no subset overlaps (I.E. they’re all Mutually Disjointed Sets).

Principal of Inclusion-Exclusion:

A related concept in Set Theory.