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