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Set Theory


									 Set Theory

By David Chester
• Was created by Georg Cantor, influenced
  by Richard Dedekind’s deep logical
• Set theory came under fire when it was
  first published in an article in August
  Crelle’s Journal in 1874.
• Leopold Kronecker was a major opponent
  to Crelle’s theories, he didn’t even believe
  in pi.
         History (continued)
• Set theory gained momentum when
  paradoxes about sets
  – such as being elements in sets being sets
    themselves gave rise to a few paradoxes
  – thanks to the excitement of this confusion, set
    theory was not abandoned.
• This led to the ZFC set theory which is one
  of the foundations of mathematics.
• A set is a collection of things that can fit in
  a category
• Basically everything can be classified into
  a set
• I.E. Letters, Numbers, Types of People
• A useful application of sets is seeing
  different sizes of infinity
• Each entry in a set is known as an
• Sets are written using curly brackets ("{"
  and "}"), with their elements listed in
    means "element of" ;

   means "not an element of"
• A union set is a combination of two or
  more sets
• It contains every element the previous sets
• Union Notation:
• If there is a similar element in each set, it
  will only be written once
           Union Application
• Set A: {8,7,4,32,23,19,30}
• Set B: {54,87,3,23,44,21}
• Union of Set A and Set B:
  –A    B: {3,4,7,8,19,21,23,30,32,44,54,87}
• An intersection set is the set of elements
  that two or more sets have in common
• It only contains the things that are the
  same, nothing else
• Intersection Notation:
•  is called the “empty set.” When two or
  more sets have nothing in common.
       Intersection Application
• Venn Diagrams are most known type of
• Set A: {65,36,87,45,46,17}
• Set B: {46,36,51,57,28,39}
• Intersection of Set A and Set B:
  –A    B: {36,46}
• A subset is a set in which every element of
  a set, basically the entire set, is part of a
  larger set.
• Subset Notation: 
• Not a Subset Notation:  
• Yet if even one element in the smaller set
  is not in the bigger set, than the smaller
  set is not a subset of the larger set.
         Subsets Application
• Set A: {14,11,56}
• Set B: {12,38,14,56,76,11}
• Since B has more elements and every
  element in A is in B, A is a subset of B:
  A B
              Sets of Numbers
• By doing sets of numbers it can be shown that they’re
  different sizes of infinity.
• Whole: Positive and Zero, No Fractions
• Integer: Positive, Negative and Zero, No fractions
• Rational: Fractions and Integers
• Irrational: Cannot be expressed as a ratio of two
  integers, i.e. pi
• Real numbers: everything above
• Imaginary Numbers: A real number times i
• All of these sets have near infinite numbers in them yet
  most of them are subsets meaning there is at least one
  set of infinity that is greater than another set, i.e.
  Rational larger than whole.
Sets of Numbers Application


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