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43-420: Fourteen is the seventh composite number . 14 is the third distinct semiprime , being the third of the form 2 × q {\displaystyle 2\times q} (where q {\displaystyle q} is a higher prime). More specifically, it is the first member of the second cluster of two discrete semiprimes (14, 15 ); the next such cluster is ( 21 , 22 ), members whose sum

86-407: A set is a collection of different things; these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. A set may have a finite number of elements or be an infinite set . There is a unique set with no elements, called the empty set ; a set with

129-457: A box containing a hat is not the same as the hat. If every element of set A is also in B , then A is described as being a subset of B , or contained in B , written A ⊆ B , or B ⊇ A . The latter notation may be read B contains A , B includes A , or B is a superset of A . The relationship between sets established by ⊆ is called inclusion or containment . Two sets are equal if they contain each other: A ⊆ B and B ⊆ A

172-516: A composite input. One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime (the factors need not be distinct, hence squares of primes are included). A composite number with three distinct prime factors is a sphenic number . In some applications, it is necessary to differentiate between composite numbers with an odd number of distinct prime factors and those with an even number of distinct prime factors. For

215-441: A set S , denoted | S | , is the number of members of S . For example, if B = {blue, white, red} , then | B | = 3 . Repeated members in roster notation are not counted, so | {blue, white, red, blue, white} | = 3 , too. More formally, two sets share the same cardinality if there exists a bijection between them. The cardinality of the empty set is zero. The list of elements of some sets

258-465: A set A to a set B is a rule that assigns to each "input" element of A an "output" that is an element of B ; more formally, a function is a special kind of relation , one that relates each element of A to exactly one element of B . A function is called An injective function is called an injection , a surjective function is called a surjection , and a bijective function is called a bijection or one-to-one correspondence . The cardinality of

301-419: A single element is a singleton . Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are equal (they are the same set). This property is called extensionality . In particular, this implies that there is only one empty set. Sets are ubiquitous in modern mathematics. Indeed, set theory , more specifically Zermelo–Fraenkel set theory , has been

344-530: Is a highly composite number (though the first two such numbers are 1 and 2). Composite numbers have also been called "rectangular numbers", but that name can also refer to the pronic numbers , numbers that are the product of two consecutive integers. Yet another way to classify composite numbers is to determine whether all prime factors are either all below or all above some fixed (prime) number. Such numbers are called smooth numbers and rough numbers , respectively. Set (mathematics) In mathematics ,

387-403: Is a subset of B , then the region representing A is completely inside the region representing B . If two sets have no elements in common, the regions do not overlap. A Venn diagram , in contrast, is a graphical representation of n sets in which the n loops divide the plane into 2 zones such that for each way of selecting some of the n sets (possibly all or none), there is a zone for

430-437: Is an element of B , this is written in shorthand as x ∈ B , which can also be read as " x belongs to B ", or " x is in B ". The statement " y is not an element of B " is written as y ∉ B , which can also be read as " y is not in B ". For example, with respect to the sets A = {1, 2, 3, 4} , B = {blue, white, red} , and F = { n | n is an integer, and 0 ≤ n ≤ 19} , The empty set (or null set )

473-407: Is an integer, and }}0\leq n\leq 19\}.} In this notation, the vertical bar "|" means "such that", and the description can be interpreted as " F is the set of all numbers n such that n is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar. Philosophy uses specific terms to classify types of definitions: If B is a set and x

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516-419: Is any subset of B (and not necessarily a proper subset), while others reserve A ⊂ B and B ⊃ A for cases where A is a proper subset of B . Examples: The empty set is a subset of every set, and every set is a subset of itself: An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If A

559-439: Is applied closure and complement operations in any possible sequence generates 14 distinct sets. This holds even if the reals are replaced by a more general topological space ; see Kuratowski's closure-complement problem . There are fourteen even numbers that cannot be expressed as the sum of two odd composite numbers : where 14 is the seventh such number. 14 is the number of equilateral triangles that are formed by

602-586: Is called squarefree . (All prime numbers and 1 are squarefree.) For example, 72 = 2 × 3 , all the prime factors are repeated, so 72 is a powerful number. 42 = 2 × 3 × 7, none of the prime factors are repeated, so 42 is squarefree. Another way to classify composite numbers is by counting the number of divisors. All composite numbers have at least three divisors. In the case of squares of primes, those divisors are { 1 , p , p 2 } {\displaystyle \{1,p,p^{2}\}} . A number n that has more divisors than any x < n

645-399: Is endless, or infinite . For example, the set N {\displaystyle \mathbb {N} } of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above are infinite. Infinite sets have infinite cardinality . Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that

688-406: Is equivalent to A = B . If A is a subset of B , but A is not equal to B , then A is called a proper subset of B . This can be written A ⊊ B . Likewise, B ⊋ A means B is a proper superset of A , i.e. B contains A , and is not equal to A . A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use A ⊂ B and B ⊃ A to mean A

731-560: Is fourteen. According to the Gospel of Matthew "there were fourteen generations in all from Abraham to David , fourteen generations from David to the exile to Babylon , and fourteen from the exile to the Messiah" ( Matthew 1, 17 ). It can also signify the Fourteen Holy Helpers . The number of pieces the body of Osiris was torn into by his fratricidal brother Set . The number 14

774-433: Is the square pyramid J 1 . {\displaystyle J_{1}.} There are a total of fourteen semi-regular polyhedra , when the pseudorhombicuboctahedron is included as a non- vertex transitive Archimedean solid (a lower class of polyhedra that follow the five Platonic solids). Fourteen possible Bravais lattices exist that fill three-dimensional space. The exceptional Lie algebra G 2

817-518: Is the fourteenth prime number, 43 . 14 has an aliquot sum of 8 , within an aliquot sequence of two composite numbers (14, 8 , 7 , 1 , 0) in the prime 7 -aliquot tree. 14 is the third companion Pell number and the fourth Catalan number . It is the lowest even n {\displaystyle n} for which the Euler totient φ ( x ) = n {\displaystyle \varphi (x)=n} has no solution, making it

860-405: Is the simplest of five such algebras, with a minimal faithful representation in fourteen dimensions. It is the automorphism group of the octonions O {\displaystyle \mathbb {O} } , and holds a compact form homeomorphic to the zero divisors with entries of unit norm in the sedenions , S {\displaystyle \mathbb {S} } . The floor of

903-400: Is the unique set that has no members. It is denoted ∅ , ∅ {\displaystyle \emptyset } , { }, ϕ , or ϕ . A singleton set is a set with exactly one element; such a set may also be called a unit set . Any such set can be written as { x }, where x is the element. The set { x } and the element x mean different things; Halmos draws the analogy that

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946-440: Is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of S with the elements of P ( S ) will leave some elements of P ( S ) unpaired. (There is never a bijection from S onto P ( S ) .) A partition of a set S is a set of nonempty subsets of S , such that every element x in S is in exactly one of these subsets. That is,

989-503: The imaginary part of the first non-trivial zero in the Riemann zeta function is 14 {\displaystyle 14} , in equivalence with its nearest integer value, from an approximation of 14.1347251417 … {\displaystyle 14.1347251417\ldots } 14 is the atomic number of silicon , and the approximate atomic weight of nitrogen . The maximum number of electrons that can fit in an f sublevel

1032-447: The sides and diagonals of a regular six-sided hexagon . In a hexagonal lattice , 14 is also the number of fixed two-dimensional triangular -celled polyiamonds with four cells. 14 is the number of elements in a regular heptagon (where there are seven vertices and edges), and the total number of diagonals between all its vertices. There are fourteen polygons that can fill a plane-vertex tiling , where five polygons tile

1075-399: The above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it. Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, Q + {\displaystyle \mathbf {Q} ^{+}} represents the set of positive rational numbers. A function (or mapping ) from

1118-400: The cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane , and indeed of any finite-dimensional Euclidean space . The continuum hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and

1161-470: The cardinality of a straight line. In 1963, Paul Cohen proved that the continuum hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice . (ZFC is the most widely-studied version of axiomatic set theory.) The power set of a set S is the set of all subsets of S . The empty set and S itself are elements of the power set of S , because these are both subsets of S . For example,

1204-414: The composite number 299 can be written as 13 × 23, and the composite number 360 can be written as 2 × 3 × 5; furthermore, this representation is unique up to the order of the factors. This fact is called the fundamental theorem of arithmetic . There are several known primality tests that can determine whether a number is prime or composite which do not necessarily reveal the factorization of

1247-422: The elements outside the union of A and B are the elements that are outside A and outside B ). The cardinality of A × B is the product of the cardinalities of A and B . This is an elementary fact when A and B are finite. When one or both are infinite, multiplication of cardinal numbers is defined to make this true. The power set of any set becomes a Boolean ring with symmetric difference as

1290-694: The elements that belong to all the selected sets and none of the others. For example, if the sets are A , B , and C , there should be a zone for the elements that are inside A and C and outside B (even if such elements do not exist). There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them. Many of these important sets are represented in mathematical texts using bold (e.g. Z {\displaystyle \mathbf {Z} } ) or blackboard bold (e.g. Z {\displaystyle \mathbb {Z} } ) typeface. These include Each of

1333-674: The first even nontotient . According to the Shapiro inequality , 14 is the least number n {\displaystyle n} such that there exist x 1 {\displaystyle x_{1}} , x 2 {\displaystyle x_{2}} , x 3 {\displaystyle x_{3}} , where: with x n + 1 = x 1 {\displaystyle x_{n+1}=x_{1}} and x n + 2 = x 2 . {\displaystyle x_{n+2}=x_{2}.} A set of real numbers to which it

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1376-550: The latter (where μ is the Möbius function and x is half the total of prime factors), while for the former However, for prime numbers, the function also returns −1 and μ ( 1 ) = 1 {\displaystyle \mu (1)=1} . For a number n with one or more repeated prime factors, If all the prime factors of a number are repeated it is called a powerful number (All perfect powers are powerful numbers). If none of its prime factors are repeated, it

1419-601: The list continues forever. For example, the set of nonnegative integers is and the set of all integers is Another way to define a set is to use a rule to determine what the elements are: Such a definition is called a semantic description . Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set F can be defined as follows: F = { n ∣ n  is an integer, and  0 ≤ n ≤ 19 } . {\displaystyle F=\{n\mid n{\text{

1462-410: The list of members can be abbreviated using an ellipsis ' ... '. For instance, the set of the first thousand positive integers may be specified in roster notation as An infinite set is a set with an infinite number of elements. If the pattern of its elements is obvious, an infinite set can be given in roster notation, with an ellipsis placed at the end of the list, or at both ends, to indicate that

1505-409: The numbers that are not prime and not a unit. E.g., the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7 –ut the integers 2 and 3 are not because each can only be divided by one and itself. The composite numbers up to 150 are: Every composite number can be written as the product of two or more (not necessarily distinct) primes. For example,

1548-621: The plane uniformly , and nine others only tile the plane alongside irregular polygons. The Klein quartic is a compact Riemann surface of genus 3 that has the largest possible automorphism group order of its kind (of order 168 ) whose fundamental domain is a regular hyperbolic 14-sided tetradecagon , with an area of 8 π {\displaystyle 8\pi } by the Gauss-Bonnet theorem . Several distinguished polyhedra in three dimensions contain fourteen faces or vertices as facets : A regular tetrahedron cell ,

1591-410: The power set of {1, 2, 3} is {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} . The power set of a set S is commonly written as P ( S ) or 2 . If S has n elements, then P ( S ) has 2 elements. For example, {1, 2, 3} has three elements, and its power set has 2 = 8 elements, as shown above. If S is infinite (whether countable or uncountable ), then P ( S )

1634-620: The set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of N {\displaystyle \mathbb {N} } are called countable sets ; these are either finite sets or countably infinite sets (sets of the same cardinality as N {\displaystyle \mathbb {N} } ); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of N {\displaystyle \mathbb {N} } are called uncountable sets . However, it can be shown that

1677-407: The simplest uniform polyhedron and Platonic solid , is made up of a total of 14 elements : 4 edges , 6 vertices, and 4 faces. 14 is also the root (non-unitary) trivial stella octangula number , where two self-dual tetrahedra are represented through figurate numbers , while also being the first non-trivial square pyramidal number (after 5 ); the simplest of the ninety-two Johnson solids

1720-472: The standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Mathematical texts commonly denote sets by capital letters in italic , such as A , B , C . A set may also be called a collection or family , especially when its elements are themselves sets. Roster or enumeration notation defines a set by listing its elements between curly brackets , separated by commas: This notation

1763-489: The subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is S . Suppose that a universal set U (a set containing all elements being discussed) has been fixed, and that A is a subset of U . Given any two sets A and B , Examples: The operations above satisfy many identities. For example, one of De Morgan's laws states that ( A ∪ B )′ = A ′ ∩ B ′ (that is,

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1806-433: Was introduced by Ernst Zermelo in 1908. In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence , a tuple , or a permutation of a set, the ordering of the terms matters). For example, {2, 4, 6} and {4, 6, 4, 2} represent the same set. For sets with many elements, especially those following an implicit pattern,

1849-409: Was regarded as connected to Šumugan and Nergal . Fourteen is: Composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime , or the unit  1, so the composite numbers are exactly

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