--- format: markdown categories: math toc: no ... # Sets *References: Most of the definitions and notation in the section are based on [^rudin] or [^meserve]* ## Basics If every element $a \in A$ is also $a \in B$, then we call A a *subset* of B and write $A \subset B$. If there are elements of B which are not elements of A, then we call A a *proper subset* of B. If $A \supset B$ and $B \supset A$ we write $A = B$; otherwise $A \neq B$. 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 symetric if $a=b$ implies $b=a$ transitive if $a=b$ and $b=c$ implies $a=c$ [^rudin]: **Principles of Mathematical Analysis (3rd ed)**, by Walter Rudin. McGraw-Hill, 1976 [^meserve]: **Fundamental Concepts of Algebra**, by Bruce Meserve.