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 diff --git a/math/sets.page b/math/sets.pageindex 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.