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