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Set Theory By David Chester History • Was created by Georg Cantor, influenced by Richard Dedekind’s deep logical thinking. • 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. Set? • 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 Notation • Each entry in a set is known as an element. • Sets are written using curly brackets ("{" and "}"), with their elements listed in between. means "element of" ; means "not an element of" Union • A union set is a combination of two or more sets • It contains every element the previous sets had • 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} Intersection • 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 sets • 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} Subsets • 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 (Selditch,4) Bibliography • http://en.wikipedia.org/wiki/Set_theory • http://www.math.niu.edu/~rusin/known- math/index/03EXX.html • http://www.geocities.com/basicmathsets/ • http://www-history.mcs.st- and.ac.uk/history/HistTopics/Beginnings_o f_set_theory.html