#### SET Theory Homework Help - K-12 Grade Level, College Level Mathematics

Introduction to Set Theory

The Set theory is a field of mathematics that studies sets. Sets are the collections of objects. However any kind of object can be collected into a set, the set theory is applied most frequently to objects which are relevant to mathematics. The set theory language can be used in the definitions of almost all mathematical objects.

The study of set theory was started by Georg Cantor and Richard Dedekind in the year 1870. Subsequent to the discovery of paradoxes in naive set theory, the number of axiom systems was proposed in the early 20th century.

Set theory is generally employed as a foundational system for mathematics, mostly in the form of Zermelo-Fraenkel set theory with the axiom of choice. Beyond the introductory role the set theory is a field of mathematics in its own right with an active research community. Contemporary research into set theory which includes a various collection of topics, varying from the structure of the real number line to the study of the consistency of large cardinals.

Table of set theory symbols

 Symbol Symbol Name Meaning / definition Example { } set a collection of elements A={3,7,9,14}, B={9,14,28} A ∩ B intersection objects that belong to set A and set B A ∩ B = {9,14} A ∪ B union objects that belong to set A or set B A ∪ B = {3,7,9,14,28} A ⊆ B subset subset has fewer elements or equal to the set {9,14,28} ⊆ {9,14,28} A ⊂ B proper subset / strict subset subset has fewer elements than the set {9,14} ⊂ {9,14,28} A ⊄ B not subset left set not a subset of right set {9,66} ⊄ {9,14,28} A ⊇ B superset set A has more elements or equal to the set B {9,14,28} ⊇ {9,14,28} A ⊃ B proper superset / strict superset set A has more elements than set B {9,14,28} ⊃ {9,14} A ? B not superset set A is not a superset of set B {9,14,28} ? {9,66} 2A power set all subsets of A ? (A) power set all subsets of A A = B equality both sets have the same members A={3,9,14}, B={3,9,14}, A=B Ac complement all the objects that do not belong to set A A \ B relative complement objects that belong to A and not to B A={3,9,14},     B={1,2,3}, A-B={9,14} A - B relative complement objects that belong to A and not to B A={3,9,14},     B={1,2,3}, A-B={9,14} A ? B symmetric difference objects that belong to A or B but not to their intersection A={3,9,14},     B={1,2,3}, A ? B={1,2,9,14} A ? B symmetric difference objects that belong to A or B but not to their intersection A={3,9,14},     B={1,2,3}, A ?B={1,2,9,14} a∈A element of set membership A={3,9,14}, 3 ∈ A x∉A not element of no set membership A={3,9,14}, 1 ∉ A (a,b) ordered pair collection of 2 elements A×B Cartesian product set of all ordered pairs from A and B |A| cardinality the number of elements of set A A={3,9,14}, |A|=3 #A cardinality the number of elements of set A A={3,9,14}, #A=3 ? aleph infinite cardinality Ø empty set Ø = { } C = {Ø} U universal set set of all possible values N0 natural numbers / whole numbers  set (with zero) N0 = {0,1,2,3,4,...} 0 ∈ N0 N1 natural numbers / whole numbers  set (without zero) N1 = {1,2,3,4,5,...} 6 ∈ N1 Z integer numbers set Z = {...-3,-2,-1,0,1,2,3,...} -6 ∈ Z Q rational numbers set Q = {x | x=a/b, a,b∈N} 2/6 ∈ Q R real numbers set R = {x | -∞ < x <∞} 6.343434 ∈ R C complex numbers set C = {z | z=a+bi, -∞

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