Axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of one object from each bin if the collection is infinite. Formally, it states that for every indexed family i ∈ I of nonempty sets there exists an indexed family i ∈ I of elements such that x i ∈ S i for every i ∈ I; the axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, such a selection can be made without invoking the axiom of choice. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. in the smallest elements are. In this case, "select the smallest number" is a choice function. If infinitely many sets were collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set.
That is, the choice function provides the set of chosen elements. However, no choice function is known for the collection of all non-empty subsets of the real numbers. In that case, the axiom of choice must be invoked. Bertrand Russell coined an analogy: for any collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection. For an infinite collection of pairs of socks, there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice. Although controversial, the axiom of choice is now used without reservation by most mathematicians, it is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice. One motivation for this use is that a number of accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy.
The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced. A choice function is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f is an element of A. With this concept, the axiom can be stated: Formally, this may be expressed as follows: ∀ X. Thus, the negation of the axiom of choice states that there exists a collection of nonempty sets that has no choice function; each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. This is not the most general situation of a Cartesian product of a family of sets, where a given set can occur more than once as a factor; the axiom of choice asserts the existence of such elements. In this article and other discussions of the Axiom of Choice the following abbreviations are common: AC – the Axiom of Choice. ZF – Zermelo–Fraenkel set theory omitting the Axiom of Choice.
ZFC – Zermelo–Fraenkel set theory, extended to include the Axiom of Choice. There are many other equivalent statements of the axiom of choice; these are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it. One variation avoids the use of choice functions by, in effect, replacing each choice function with its range. Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains one element in common with each of the sets in X; this guarantees for any partition of a set X the existence of a subset C of X containing one element from each part of the partition. Another equivalent axiom only considers collections X that are powersets of other sets: For any set A, the power set of A has a choice function. Authors who use this formulation speak of the choice function on A, but be advised that this is a different notion of choice function, its domain is the powerset of A, and
Jean van Heijenoort
Jean Louis Maxime van Heijenoort was a historian of mathematical logic. He was a personal secretary to Leon Trotsky from 1932 to 1939, from until 1947, an American Trotskyist activist. Van Heijenoort was born in France, his family's financial circumstances were difficult as his Dutch immigrant father died when van Heijenoort was two. He acquired a powerful traditional French formal education, to which his French writings attest. Although he became a naturalized American citizen, he visited France twice a year from 1958 until his death, remained attached to his French extended family and friends. In 1932, he joined the Communist League. Soon thereafter, the exiled Trotsky hired van Heijenoort as a secretary and bodyguard because of his fluency in French, Russian and English, thus began seven years in Trotsky's household, during which he served as an all-purpose translator, helping Trotsky write several books and keep up an extensive intellectual and political correspondence in several languages.
In 1939, van Heijenoort moved to New York City to be with his second wife, Beatrice "Bunny" Guyer, where he worked for the Socialist Workers Party and wrote a number of articles for the American Trotskyist press and other radical outlets. He was elected to the secretariat of the Fourth International in 1940 but resigned when Felix Morrow and Albert Goldman, with whom he had sided, were expelled from the SWP. Goldman subsequently went on to join the US Workers Party but Morrow joined no other party/grouping. In 1947, he too was expelled from the SWP. In 1948, he published an article, called "A Century's Balance Sheet" in which he criticized that part of Marxism which saw the "proletariat" as the revolutionary class, he continued to hold other parts of Marxism as true. Van Heijenoort was spared the ordeal of McCarthyism because everything he published in Trotskyist organs appeared under one or other of more than a dozen pen names. Moreover, Feferman states that van Heijenoort the logician was quite reticent about his Trotskyist youth, did not discuss politics.
In the last decade of his life he contributed to the history of the Trotskyist movement by writing the monograph With Trotsky in Exile, editing a volume of Trotsky's correspondence, advising and working with the archivists at the Houghton Library in Harvard University, which holds many of Trotsky's papers from his years in exile. After completing a Ph. D. in mathematics at New York University in 1949 under the supervision of J. J. Stoker, he taught mathematics there but evolved into a logician and philosopher of mathematics, in good part because of the influence of Georg Kreisel, he began teaching philosophy, first part-time at Columbia University full-time at Brandeis University, 1965-77. He spent much of his last decade at Stanford University and editing eight books, including parts of the Collected Works of Kurt Gödel; the Source Book the most important book published on the history of logic and of the foundations of mathematics, is an anthology of translations. It begins with the first complete translation of Frege's 1879 Begriffsschrift, followed by 45 important short pieces on mathematical logic and axiomatic set theory published between 1889 and 1931.
The anthology ends with Gödel's landmark paper on the incompletability of Peano arithmetic. For more information on the period covered by this anthology, see Grattan-Guinness. Nearly all the content of the Source Book was difficult to access in all but the best North American university libraries, all but four pieces had to be translated from one of six continental European languages; when possible, the author of the original text was asked to review the translation of his work, suggest corrections and amendments. Each piece included editorial footnotes, all references were combined into one list, many misprints and errors in the originals were corrected. Important are the remarkable introductions to each translation, most written by van Heijenoort himself. A few were written by Burton Dreben; the Source Book did much to advance the view that modern logic begins with, builds on, the Begriffsschrift. Grattan-Guinness argues that this perspective on the history of logic is mistaken, because Frege employed an idiosyncratic notation and was far less read than, Peano.
Van Heijenoort is cited by those who prefer the alternative model theoretic stance on logic and mathematics. Much of the history of that stance, whose leading lights include George Boole, Charles Sanders Peirce, Ernst Schröder, Leopold Löwenheim, Thoralf Skolem, Alfred Tarski, Jaakko Hintikka, is covered in Brady; the Source Book underrated the algebraic logic of De Morgan, Boole and Schröder, but devoted more pages to Skolem than to anyone other than Frege, included Löwenheim, the founding paper on model theory. Van Heijenoort had children with two of his four wives. While living with Trotsky in Coyoacán, now a neighborhood of Mexico City, van Heijenoort's first wife left him after clashing with Trotsky's spouse. Van Heijenoort was one of Frida Kahlo's lovers. Having parted company with Trotsky in 1939 for personal reasons, van Heijenoort was innocent of all circumstances leading to Trotsky's 1940 murder. Van Heijenoort himself was murdered, in Mexico City 46 years later
A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves circles, each representing a set; the points inside a curve labelled S represent elements of the set S, while points outside the boundary represent elements not in the set S. This lends to read visualizations. In Venn diagrams the curves are overlapped in every possible way, showing all possible relations between the sets, they are thus a special case of Euler diagrams, which do not show all relations. Venn diagrams were conceived around 1880 by John Venn, they are used to teach elementary set theory, as well as illustrate simple set relationships in probability, statistics and computer science. A Venn diagram in which the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram.
This example involves A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures; the blue circle, set B, represents the living creatures. Each separate type of creature can be imagined as a point somewhere in the diagram. Living creatures that both can fly and have two legs—for example, parrots—are in both sets, so they correspond to points in the region where the blue and orange circles overlap, it is important to note that this overlapping region would only contain those elements that are members of both set A and are members of set B Humans and penguins are bipedal, so are in the orange circle, but since they cannot fly they appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes have six legs, fly, so the point for mosquitoes is in the part of the blue circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly would all be represented by points outside both circles.
The combined region of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that can fly; the region in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles. Venn diagrams were introduced in 1880 by John Venn in a paper entitled On the Diagrammatic and Mechanical Representation of Propositions and Reasonings in the "Philosophical Magazine and Journal of Science", about the different ways to represent propositions by diagrams; the use of these types of diagrams in formal logic, according to Frank Ruskey and Mark Weston, is "not an easy history to trace, but it is certain that the diagrams that are popularly associated with Venn, in fact, originated much earlier. They are rightly associated with Venn, because he comprehensively surveyed and formalized their usage, was the first to generalize them".
Venn himself did not use the term "Venn diagram" and referred to his invention as "Eulerian Circles". For example, in the opening sentence of his 1880 article Venn writes, "Schemes of diagrammatic representation have been so familiarly introduced into logical treatises during the last century or so, that many readers those who have made no professional study of logic, may be supposed to be acquainted with the general nature and object of such devices. Of these schemes one only, viz. that called'Eulerian circles,' has met with any general acceptance..." Lewis Carroll includes "Venn's Method of Diagrams" as well as "Euler's Method of Diagrams" in an "Appendix, Addressed to Teachers" of his book "Symbolic Logic". The term "Venn diagram" was used by Clarence Irving Lewis in 1918, in his book "A Survey of Symbolic Logic". Venn diagrams are similar to Euler diagrams, which were invented by Leonhard Euler in the 18th century. M. E. Baron has noted that Leibniz in the 17th century produced similar diagrams before Euler, but much of it was unpublished.
She observes earlier Euler-like diagrams by Ramon Llull in the 13th Century. In the 20th century, Venn diagrams were further developed. D. W. Henderson showed in 1963 that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was a prime number, he showed that such symmetric Venn diagrams exist when n is five or seven. In 2002 Peter Hamburger found symmetric Venn diagrams for n = 11 and in 2003, Griggs and Savage showed that symmetric Venn diagrams exist for all other primes, thus rotationally symmetric Venn diagrams exist. Venn diagrams and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement in the 1960s. Since they have been adopted in the curriculum of other fields such as reading. A Venn diagram is constructed with a collection of simple closed curves drawn in a plane. According to Lewis, the "principle of these diagrams is that classes be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram.
That is, the diagram leaves room for any possible relation
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set. Writing A = means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example, are subsets of A. Sets can themselves be elements. For example, consider the set B =; the elements of B are not 1, 2, 3, 4. Rather, there are only three elements of B, namely the numbers 1 and 2, the set; the elements of a set can be anything. For example, C =, is the set whose elements are the colors red and blue; the relation "is an element of" called set membership, is denoted by the symbol " ∈ ". Writing x ∈ A means that "x is an element of A". Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A" and "x lies in A"; the expressions "A includes x" and "A contains x" are used to mean set membership, however some authors use them to mean instead "x is a subset of A". Logician George Boolos urged that "contains" be used for membership only and "includes" for the subset relation only.
For the relation ∈, the converse relation ∈T may be written A ∋ x, meaning "A contains x". The negation of set membership is denoted by the symbol "∉". Writing x ∉ A means that "x is not an element of A"; the symbol ∈ was first used by Giuseppe Peano 1889 in his work Arithmetices principia, nova methodo exposita. Here he wrote on page X: Signum ∈ significat est. Ita a ∈ b legitur a est quoddam b. So a ∈ b is read; every relation R: U → V is subject to two involutions: complementation R → R ¯ and conversion RT: V → U. The relation ∈ has for its domain a universal set U, has the power set P for its codomain or range; the complementary relation ∈ ¯ = ∉ expresses the opposite of ∈. An element x ∈ U may have x ∉ A, in which case x ∈ U \ A, the complement of A in U; the converse relation ∈ T = ∋ swaps the domain and range with ∈. For any A in P, A ∋ x is true when x ∈ A; the number of elements in a particular set is a property known as cardinality. In the above examples the cardinality of the set A is 4, while the cardinality of either of the sets B and C is 3.
An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of positive integers. Using the sets defined above, namely A =, B = and C =: 2 ∈ A ∈ B 3,4 ∉ B is a member of B Yellow ∉ C The cardinality of D = is finite and equal to 5; the cardinality of P = is infinite. Halmos, Paul R. Naive Set Theory, Undergraduate Texts in Mathematics, NY: Springer-Verlag, ISBN 0-387-90092-6 - "Naive" means that it is not axiomatized, not that it is silly or easy. Jech, Thomas, "Set Theory", Stanford Encyclopedia of Philosophy Suppes, Axiomatic Set Theory, NY: Dover Publications, Inc. ISBN 0-486-61630-4 - Both the notion of set, membership or element-hood, the axiom of extension, the axiom of separation, the union axiom are needed for a more thorough understanding of "set element". Weisstein, Eric W. "Element". MathWorld
In set theory, a Cartesian product is a mathematical operation that returns a set from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs where a ∈ A and b ∈ B. Products can be specified using e.g.. A × B =. A table can be created by taking the Cartesian product of a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form. More a Cartesian product of n sets known as an n-fold Cartesian product, can be represented by an array of n dimensions, where each element is an n-tuple. An ordered pair is a couple; the Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, further generalized in terms of direct product. An illustrative example is the standard 52-card deck; the standard playing card ranks form a 13-element set. The card suits form a four-element set; the Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.
Ranks × Suits returns a set of the form. Suits × Ranks returns a set of the form. Both sets are distinct disjoint; the main historical example is the Cartesian plane in analytic geometry. In order to represent geometrical shapes in a numerical way and extract numerical information from shapes' numerical representations, René Descartes assigned to each point in the plane a pair of real numbers, called its coordinates; such a pair's first and second components are called its x and y coordinates, respectively. The set of all such pairs is thus assigned to the set of all points in the plane. A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair; the most common definition of ordered pairs, the Kuratowski definition, is =. Under this definition, is an element of P, X × Y is a subset of that set, where P represents the power set operator. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, power set, specification.
Since functions are defined as a special case of relations, relations are defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is prior to most other definitions. Let A, B, C, D be sets; the Cartesian product A × B is not commutative, A × B ≠ B × A, because the ordered pairs are reversed unless at least one of the following conditions is satisfied: A is equal to B, or A or B is the empty set. For example: A =. × C ≠ A × If for example A = × A = ≠ = A ×. The Cartesian product behaves nicely with respect to intersections. × = ∩. × ≠ ∪ In fact, we have that: ∪ = ∪ ∪ [ ( B
Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of mathematical logic, probability theory, operator theory, ergodic theory, functional analysis. He was recognized as a great mathematical expositor, he has been described as one of The Martians. Born in Hungary into a Jewish family, Halmos arrived in the U. S. at 13 years of age. He obtained his B. A. from the University of Illinois, majoring in mathematics, but fulfilling the requirements for both a math and philosophy degree. He took only three years to obtain the degree, was only 19 when he graduated, he began a Ph. D. in philosophy, still at the Champaign-Urbana campus. Joseph L. Doob supervised his dissertation, titled Invariants of Certain Stochastic Transformations: The Mathematical Theory of Gambling Systems. Shortly after his graduation, Halmos left for the Institute for Advanced Study, lacking both job and grant money. Six months he was working under John von Neumann, which proved a decisive experience.
While at the Institute, Halmos wrote his first book, Finite Dimensional Vector Spaces, which established his reputation as a fine expositor of mathematics. Halmos taught at Syracuse University, the University of Chicago, the University of Michigan, the University of California at Santa Barbara, the University of Hawaii, Indiana University. From his 1985 retirement from Indiana until his death, he was affiliated with the Mathematics department at Santa Clara University. In a series of papers reprinted in his 1962 Algebraic Logic, Halmos devised polyadic algebras, an algebraic version of first-order logic differing from the better known cylindric algebras of Alfred Tarski and his students. An elementary version of polyadic algebra is described in monadic Boolean algebra. In addition to his original contributions to mathematics, Halmos was an unusually clear and engaging expositor of university mathematics, he won the Lester R. Ford Award in 1971 and again in 1977. Halmos chaired the American Mathematical Society committee that wrote the AMS style guide for academic mathematics, published in 1973.
In 1983, he received the AMS's Steele Prize for exposition. In the American Scientist 56: 375–389, Halmos argued that mathematics is a creative art, that mathematicians should be seen as artists, not number crunchers, he discussed the division of the field into mathology and mathophysics, further arguing that mathematicians and painters think and work in related ways. Halmos's 1985 "automathography" I Want to Be a Mathematician is an account of what it was like to be an academic mathematician in 20th century America, he called the book "automathography" rather than "autobiography", because its focus is entirely on his life as a mathematician, not his personal life. The book contains the following quote on Halmos' view of what doing mathematics means: Don't just read it. Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
In these memoirs, Halmos claims to have invented the "iff" notation for the words "if and only if" and to have been the first to use the "tombstone" notation to signify the end of a proof, this is agreed to be the case. The tombstone symbol ∎ is sometimes called a halmos. In 2005, Halmos and his wife Virginia funded the Euler Book Prize, an annual award given by the Mathematical Association of America for a book, to improve the view of mathematics among the public; the first prize was given in 2007, the 300th anniversary of Leonhard Euler's birth, to John Derbyshire for his book about Bernhard Riemann and the Riemann hypothesis: Prime Obsession. 1942. Finite-Dimensional Vector Spaces. Springer-Verlag. 1950. Measure Theory. Springer Verlag. 1951. Introduction to Hilbert Space and the Theory of Spectral Multiplicity. Chelsea. 1956. Lectures on Ergodic Theory. Chelsea. 1960. Naive Set Theory. Springer Verlag. 1962. Algebraic Logic. Chelsea. 1963. Lectures on Boolean Algebras. Van Nostrand. 1967. A Hilbert Space Problem Book.
Springer-Verlag. 1973.. How to Write Mathematics. American Mathematical Society. 1978.. Bounded Integral Operators on L² Spaces. Springer Verlag 1985. I Want to Be a Mathematician. Springer-Verlag. 1987. I Have a Photographic Memory. Mathematical Association of America. 1991. Problems for Mathematicians and Old, Dolciani Mathematical Expositions, Mathematical Association of America. 1996. Linear Algebra Problem Book, Dolciani Mathematical Expositions, Mathematical Association of America. 1998.. Logic as Algebra, Dolciani Mathematical Expositions No. 21, Mathematical Association of America. 2009. Introduction to Boolean Algebras, Springer. Criticism of non-standard analysis The Martians J. H. Ewing. Paul Halmos: Celebrating 50 Years of Mathematics. Springer-Verlag. ISBN 0-387-97509-8. OCLC 22859036. Includes a bibliography of Halmos's writings through 1991. John Ewing. "Paul Halmos: In His Own Words". Notices of the American Mathematical Society. 54: 1136–1144. Retrieved 2008-01-15. Paul Halmos. I want to be a Mathematician: An Automathography.
Springer-Verlag. ISBN 0-387-96470-3. OCLC 230812318. O'Connor, John J..
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most to objects that are relevant to mathematics; the language of set theory can be used to define nearly all mathematical objects. The modern study of set theory was initiated by Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known. Set theory is employed as a foundational system for mathematics in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
Mathematical topics emerge and evolve through interactions among many researchers. Set theory, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874 and was motivated by Cantor's work in real analysis. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Cantor's work polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, the "infinity of infinities" resulting from the power set operation.
This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia. The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics. In 1906 English readers gained the book Theory of Sets of Points by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.
The momentum of set theory was such. The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most used set of axioms for set theory; the work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology—category theory is thought to be a preferred foundation. Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member of A, the notation o. Since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the subset relation called set inclusion. If all the members of set A are members of set B A is a subset of B, denoted A ⊆ B. For example, is a subset of, so is but is not; as insinuated from this definition, a set is a subset of itself.
For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B. Note that 1, 2, 3 are members of the set but are not subsets of it. Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The: Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both; the union of and is the set. Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B; the intersection of and is the set. Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A; the set difference \ is, conversely, the set difference \ is. When A is a subset of U, the set difference U \ A is called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U \ A if U is a universal set as in the study of Venn diagrams.
Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is