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unserer ansehauung oder unseres denkens welche die elemente der menge genannt werden zu einem ganzen 16 a set is a collection into a whole of definite distinct objects of our perception or of our thought which are called elements of the set citation needed georg cantor in h baumann et al lehrbuch der mathematik für wirtschaftswissenschaften 1975 chapter 2 grundbegriffe der mengenlehre note on consistency edit it does not follow from this definition how sets can be formed and what operations on sets again will produce a set the term well defined in well defined collection of objects cannot by itself guarantee the consistency and unambiguity of what exactly constitutes and what does not constitute a set attempting to achieve this would be the realm of axiomatic set theory or of axiomatic class theory the problem in this context with informally formulated set theories not derived from and implying any particular axiomatic theory is that there may be several widely differing formalized versions that have both different sets and different rules for how new sets may be formed that all conform to the original informal definition for example cantor s verbatim definition allows for considerable freedom in what constitutes a set on the other hand it is unlikely that cantor was particularly interested in sets containing cats and dogs but rather only in sets containing purely mathematical objects an example of such a class of sets could be the von neumann universe but even when fixing the class of sets under consideration it is not always clear which rules for set formation are allowed without introducing paradoxes for the purpose of fixing the discussion below the term well defined should instead be interpreted as an intention with either implicit or explicit rules axioms or definitions to rule out inconsistencies the purpose is to keep the often deep and difficult issues of consistency away from the usually simpler context at hand an explicit ruling out of all conceivable inconsistencies paradoxes cannot be achieved for an axiomatic set theory anyway due to gödel s second incompleteness theorem so this does not at all hamper the utility of naive set theory as compared to axiomatic set theory in the simple contexts considered below it merely simplifies the discussion consistency is henceforth taken for granted unless explicitly mentioned membership edit first usage of the symbol ϵ in the work arithmetices principia nova methodo exposita by giuseppe peano if x is a member of a set a then it is also said that x belongs to a or that x is in a this is denoted by x a the symbol is a derivation from the lowercase greek letter epsilon ε introduced by giuseppe peano in 1889 and is the first letter of the word ἐστί means is the symbol is often used to write x a meaning x is not in a equality edit two sets a and b are defined to be equal when they have precisely the same elements that is if every element of a is an element of b and every element of b is an element of a see axiom of extensionality thus a set is completely determined by its elements the description is immaterial for example the set with elements 2 3 and 5 is equal to the set of all prime numbers less than 6 if the sets a and b are equal this is denoted symbolically as a b as usual empty set edit the empty set denoted as displaystyle varnothing and sometimes displaystyle is a set with no members at all because a set is determined completely by its elements there can be only one empty set see axiom of empty set 17 although the empty set has no members it can be a member of other sets thus displaystyle varnothing neq varnothing because the former has no members and the latter has one member 18 specifying sets edit the simplest way to describe a set is to list its elements between curly braces known as defining a set extensionally thus 1 2 denotes the set whose only elements are 1 and 2 see axiom of pairing note the following points the order of elements is immaterial for example 1 2 2 1 repetition multiplicity of elements is irrelevant for example 1 2 2 1 1 1 2 1 2 these are consequences of the definition of equality in the previous section this notation can be informally abused by saying something like dogs to indicate the set of all dogs but this example would usually be read by mathematicians as the set containing the single element dogs an extreme but correct example of this notation is which denotes the empty set the notation x p x or sometimes x p x is used to denote the set containing all objects for which the condition p holds known as defining a set intensionally for example x x r denotes the set of real numbers x x has blonde hair denotes the set of everything with blonde hair this notation is called set builder notation or set comprehension particularly in the context of functional programming some variants of set builder notation are x a p x denotes the set of all x that are already members of a such that the condition p holds for x for example if z is the set of integers then x z x is even is the set of all even integers see axiom of specification f x x a denotes the set of all objects obtained by putting members of the set a into the formula f for example 2 x x z is again the set of all even integers see axiom of replacement f x p x is the most general form of set builder notation for example x s owner x is a dog is the set of all dog owners subsets edit given two sets a and b a is a subset of b if every element of a is also an element of b in particular each set b is a subset of itself a subset of b that is not equal to b is called a proper subset if a is a subset of b then one can also say that b is a superset of a that a is contained in b or that b contains a in symbols a b means that a is a subset of b and b a means that b is a superset of a some authors use the symbols and for subsets and others use these symbols only for proper subsets for clarity one can explicitly use the symbols and to indicate non equality as an illustration let r be the set of real numbers let z be the set of integers let o be the set of odd integers and let p be the set of current or former u s presidents then o is a subset of z z is a subset of r and hence o is a subset of r where in all cases subset may even be read as proper subset not all sets are comparable in this way for example it is not the case either that r is a subset of p nor that p is a subset of r it follows immediately from the definition of equality of sets above that given two sets a and b a b if and only if a b and b a in fact this is often given as the definition of equality usually when trying to prove that two sets are equal one aims to show these two inclusions the empty set is a subset of every set the statement that all elements of the empty set are also members of any set a is vacuously true the set of all subsets of a given set a is called the power set of a and is denoted by 2 a displaystyle 2 a or p a displaystyle p a the p is sometimes in a script font a displaystyle wp a if the set a has n elements then p a displaystyle p a will have 2 n displaystyle 2 n elements universal sets and absolute complements edit in certain contexts one may consider all sets under consideration as being subsets of some given universal set for instance when investigating properties of the real numbers r and subsets of r r may be taken as the universal set a true universal set is not included in standard set theory see paradoxes below but is included in some non standard set theories given a universal set u and a subset a of u the complement of a in u is defined as a c x u x a in other words a c a complement sometimes simply a a prime is the set of all members of u which are not members of a thus with r z and o defined as in the section on subsets if z is the universal set then o c is the set of even integers while if r is the universal set then o c is the set of all real numbers that are either even integers or not integers at all unions intersections and relative complements edit given two sets a and b their union is the set consisting of all objects which are elements of a or of b or of both see axiom of union it is denoted by a b the intersection of a and b is the set of all objects which are both in a and in b it is denoted by a b finally the relative complement of b relative to a also known as the set theoretic difference of a and b is the set of all objects that belong to a but not to b it is written as a b or a b symbolically these are respectively a b x x a x b a b x x a x b x a x b x b x a a b x x a x b x a x b the set b doesn t have to be a subset of a for a b to make sense this is the difference between the relative complement and the absolute complement a c u a from the previous section to illustrate these ideas let a be the set of left handed people and let b be the set of people with blond hair then a b is the set of all left handed blond haired people while a b is the set of all people who are left handed or blond haired or both a b on the other hand is the set of all people that are left handed but not blond haired while b a is the set of all people who have blond hair but aren t left handed now let e be the set of all human beings and let f be the set of all living things over 1000 years old what is e f in this case no living human being is over 1000 years old so e f must be the empty set for any set a the power set p a displaystyle p a is a boolean algebra under the operations of union and intersection ordered pairs and cartesian products edit intuitively an ordered pair is simply a collection of two objects such that one can be distinguished as the first element and the other as the second element and having the fundamental property that two ordered pairs are equal if and only if their first elements are equal and their second elements are equal formally an ordered pair with first coordinate a and second coordinate b usually denoted by a b can be defined as the set a a b displaystyle a a b it follows that two ordered pairs a b and c d are equal if and only if a c and b d alternatively an ordered pair can be formally thought of as a set a b with a total order the notation a b is also used to denote an open interval on the real number line but the context should make it clear which meaning is intended otherwise the notation a b may be used to denote the open interval whereas a b is used for the ordered pair if a and b are sets then the cartesian product or simply product is defined to be a b a b a a and b b that is a b is the set of all ordered pairs whose first coordinate is an element of a and whose second coordinate is an element of b this definition may be extended to a set a b c of ordered triples and more generally to sets of ordered n tuples for any positive integer n it is even possible to define infinite cartesian products but this requires a more recondite definition of the product cartesian products were first developed by rené descartes in the context of analytic geometry if r denotes the set of all real numbers then r 2 r r represents the euclidean plane and r 3 r r r represents three dimensional euclidean space some important sets edit there are some ubiquitous sets for which the notation is almost universal some of these are listed below in the list a b and c refer to natural numbers and r and s are real numbers natural numbers are used for counting a blackboard bold capital n n displaystyle mathbb n often represents this set integers appear as solutions for x in equations like x a b a blackboard bold capital z z displaystyle mathbb z often represents this set from the german zahlen meaning numbers rational numbers appear as solutions to equations like a bx c a blackboard bold capital q q displaystyle mathbb q often represents this set for quotient because r is used for the set of real numbers algebraic numbers appear as solutions to polynomial equations with integer coefficients and may involve radicals including i 1 displaystyle i sqrt 1 and certain other irrational numbers a q with an overline q displaystyle overline mathbb q often represents this set the overline denotes the operation of algebraic closure real numbers represent the real line and include all numbers that can be approximated by rationals these numbers may be rational or algebraic but may also be transcendental numbers which cannot appear as solutions to polynomial equations with rational coefficients a blackboard bold capital r r displaystyle mathbb r often represents this set complex numbers are sums of a real and an imaginary number r s i displaystyle r s i here either r displaystyle r or s displaystyle s or both can be zero thus the set of real numbers and the set of strictly imaginary numbers are subsets of the set of complex numbers which form an algebraic closure for the set of real numbers meaning that every polynomial with coefficients in r displaystyle mathbb r has at least one root in this set a blackboard bold capital c c displaystyle mathbb c often represents this set note that since a number r s i displaystyle r s i can be identified with a point r s displaystyle r s in the plane c displaystyle mathbb c is basically the same as the cartesian product r r displaystyle mathbb r times mathbb r the same meaning that any point in one determines a unique point in the other and for the result of calculations it doesn t matter which one is used for the calculation as long as multiplication rule is appropriate for c displaystyle mathbb c paradoxes in early set theory edit main article paradox the unrestricted formation principle of sets referred to as the axiom schema of unrestricted comprehension if p is a property then there exists a set y x p x 19 is the source of several early appearing paradoxes y x x is an ordinal led in the year 1897 to the burali forti paradox the first published antinomy y x x is a cardinal produced cantor s paradox in 1897 9 y x yielded cantor s second antinomy in the year 1899 11 here the property p is true for all x whatever x may be so y would be a universal set containing everything y x x x i e the set of all sets that do not contain themselves as elements gave russell s paradox in 1902 if the axiom schema of unrestricted comprehension is weakened to the axiom schema of specification or axiom schema of separation if p is a property then for any set x there exists a set y x x p x 19 then all the above paradoxes disappear 19 there is a corollary with the axiom schema of separation as an axiom of the theory it follows as a theorem of the theory the set of all sets does not exist or more spectacularly halmos phrasing 20 there is no universe proof suppose that it exists and call it u now apply the axiom schema of separation with x u and for p x use x x this leads to russell s paradox again hence u cannot exist in this theory 19 related to the above constructions is formation of the set y x x x where the statement following the implication certainly is false it follows from the definition of ...