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diff --git a/math/sets.page b/math/sets.page index 42eb831..7b464ed 100644 --- a/math/sets.page +++ b/math/sets.page @@ -1,25 +1,30 @@ -==================== -Sets -==================== +--- +format: markdown +categories: math +toc: no +... -.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_ +# Sets -Basics -============= -If every element :latex:`$a \in A$` is also :latex:`$a \in B$`, then we call -A a *subset* of B and write :latex:`$A \subset B$`. If there are elements of B +*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 :latex:`$A \supset B$` and :latex:`$B \supset A$` we write :latex:`$A = B$`; -otherwise :latex:`$A \neq B$`. +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 + 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 @@ -28,20 +33,20 @@ elements of the set is unique from the results from all other pairs of elements. Some equivalence relations are -:latex:`$\identity$` (NOTE: = with three lines) (*identity*); -:latex:`$\congruence$` (NOTE: = with tilde on top) (*congruence*; eg of +$\identity$ (NOTE: = with three lines) (*identity*); +$\congruence$ (NOTE: = with tilde on top) (*congruence*; eg of geometric figures); and -:latex:`$~$` (NOTE: tilde) (*similarity*; eg of geometric figures). +$~$ (NOTE: tilde) (*similarity*; eg of geometric figures). Some properties of equivalence relations are -*reflexive* - if :latex:`$a=a$` is true for all a -*symetric* - if :latex:`$a=b$` implies :latex:`$b=a$` -*transitive* - if :latex:`$a=b$` and :latex:`$b=c$` implies :latex:`$a=c$` +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)`:title:, by Walter Rudin. McGraw-Hill, 1976 +[^rudin]: **Principles of Mathematical Analysis (3rd ed)**, by Walter Rudin. McGraw-Hill, 1976 -.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve. +[^meserve]: **Fundamental Concepts of Algebra**, by Bruce Meserve. |