# math/sets

# Sets

*References: Most of the definitions and notation in the section are based on ^{1} or ^{2}*

## Basics

If every element *a* ∈ *A* is also *a* ∈ *B*, then we call A a *subset* of B and write *A* ⊂ *B*. If there are elements of B which are not elements of A, then we call A a *proper subset* of B.

If *A* ⊃ *B* and *B* ⊃ *A* we write *A* = *B*; otherwise *A* ≠ *B*.

The null or empty set, which has no elements, is a subset of all others.

A relation on a space of sets S is something that can be definted as either true or false (holding or not holding) for any binary pair in S.

# Binary Operators

Binary operators defined on a set apply to any two elements of that set; order may or may not be important. A set is *closed* with regards to a binary operator if it contains the result of the binary operator. A set is *uniquely defined* with regards to a binary operator if the result of the operator on two elements of the set is unique from the results from all other pairs of elements.

Some equivalence relations are $\identity$ (NOTE: = with three lines) (*identity*); $\congruence$ (NOTE: = with tilde on top) (*congruence*; eg of geometric figures); and (NOTE: tilde) (*similarity*; eg of geometric figures).

Some properties of equivalence relations are

- reflexive
- if
*a*=*a*is true for all a - symmetric
- if
*a*=*b*implies*b*=*a* - transitive
- if
*a*=*b*and*b*=*c*implies*a*=*c*