If A is a null set and P A denotes power set of A then the number of elements of P P P A is

Main article: Null sign

Common notations for the empty set include "{}", " ∅ {\displaystyle \emptyset }  ", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Danish and Norwegian alphabets.[2] In the past, "0" was occasionally used as a symbol for the empty set, but this is now considered to be an improper use of notation.[3]

The symbol ∅ is available at Unicode point U+2205.[4] It can be coded in HTML as ∅ and as ∅. It can be coded in LaTeX as \varnothing. The symbol ∅ {\displaystyle \emptyset }   is coded in LaTeX as \emptyset.

When writing in languages such as Danish and Norwegian, where the empty set character may be confused with the alphabetic letter Ø (as when using the symbol in linguistics), the Unicode character U+29B0 REVERSED EMPTY SET ⦰ may be used instead.[5]

In standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements. As a result, there can be only one set with no elements, hence the usage of "the empty set" rather than "an empty set".

The empty set has the following properties:

• Its only subset is the empty set itself: ∀ A : A ⊆ ∅ ⇒ A = ∅ {\displaystyle \forall A:A\subseteq \varnothing \Rightarrow A=\varnothing }  
• The power set of the empty set is the set containing only the empty set: 2 ∅ = { ∅ } {\displaystyle 2^{\varnothing }=\{\varnothing \}}  
• The number of elements of the empty set (i.e., its cardinality) is zero: | ∅ | = 0 {\displaystyle \mathrm {|} \varnothing \mathrm {|} =0}  

For any set A:

• The empty set is a subset of A: ∀ A : ∅ ⊆ A {\displaystyle \forall A:\varnothing \subseteq A}  
• The union of A with the empty set is A: ∀ A : A ∪ ∅ = A {\displaystyle \forall A:A\cup \varnothing =A}  
• The intersection of A with the empty set is the empty set: ∀ A : A ∩ ∅ = ∅ {\displaystyle \forall A:A\cap \varnothing =\varnothing }  
• The Cartesian product of A and the empty set is the empty set: ∀ A : A × ∅ = ∅ {\displaystyle \forall A:A\times \varnothing =\varnothing }  

For any property P:

• For every element of ∅ {\displaystyle \varnothing }  , the property P holds (vacuous truth).
• There is no element of ∅ {\displaystyle \varnothing }   for which the property P holds.

Conversely, if for some property P and some set V, the following two statements hold:

• For every element of V the property P holds
• There is no element of V for which the property P holds

then V = ∅ . {\displaystyle V=\varnothing .}  

By the definition of subset, the empty set is a subset of any set A. That is, every element x of ∅ {\displaystyle \varnothing }   belongs to A. Indeed, if it were not true that every element of ∅ {\displaystyle \varnothing }   is in A, then there would be at least one element of ∅ {\displaystyle \varnothing }   that is not present in A. Since there are no elements of ∅ {\displaystyle \varnothing }   at all, there is no element of ∅ {\displaystyle \varnothing }   that is not in A. Any statement that begins "for every element of ∅ {\displaystyle \varnothing }  " is not making any substantive claim; it is a vacuous truth. This is often paraphrased as "everything is true of the elements of the empty set."

In the usual set-theoretic definition of natural numbers, zero is modelled by the empty set.

Operations on the empty set

When speaking of the sum of the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set is zero. The reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set should be considered to be one (see empty product), since one is the identity element for multiplication.

A derangement is a permutation of a set without fixed points. The empty set can be considered a derangement of itself, because it has only one permutation ( 0 ! = 1 {\displaystyle 0!=1}  ), and it is vacuously true that no element (of the empty set) can be found that retains its original position.

Since the empty set has no member when it is considered as a subset of any ordered set, every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower bound for the empty set.[6] When considered as a subset of the extended reals formed by adding two "numbers" or "points" to the real numbers (namely negative infinity, denoted − ∞ , {\displaystyle -\infty \!\,,}   which is defined to be less than every other extended real number, and positive infinity, denoted + ∞ , {\displaystyle +\infty \!\,,}   which is defined to be greater than every other extended real number), we have that:

sup ∅ = min ( { − ∞ , + ∞ } ∪ R ) = − ∞ , {\displaystyle \sup \varnothing =\min(\{-\infty ,+\infty \}\cup \mathbb {R} )=-\infty ,}

inf ∅ = max ( { − ∞ , + ∞ } ∪ R ) = + ∞ . {\displaystyle \inf \varnothing =\max(\{-\infty ,+\infty \}\cup \mathbb {R} )=+\infty .}

That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for the minimum and infimum operators.

Topology

In any topological space X, the empty set is open by definition, as is X. Since the complement of an open set is closed and the empty set and X are complements of each other, the empty set is also closed, making it a clopen set. Moreover, the empty set is compact by the fact that every finite set is compact.

The closure of the empty set is empty. This is known as "preservation of nullary unions."

Category theory

If A {\displaystyle A}   is a set, then there exists precisely one function f {\displaystyle f}   from ∅ {\displaystyle \varnothing }   to A , {\displaystyle A,}   the empty function. As a result, the empty set is the unique initial object of the category of sets and functions.

The empty set can be turned into a topological space, called the empty space, in just one way: by defining the empty set to be open. This empty topological space is the unique initial object in the category of topological spaces with continuous maps. In fact, it is a strict initial object: only the empty set has a function to the empty set.

Set theory

In the von Neumann construction of the ordinals, 0 is defined as the empty set, and the successor of an ordinal is defined as S ( α ) = α ∪ { α } {\displaystyle S(\alpha )=\alpha \cup \{\alpha \}}  . Thus, we have 0 = ∅ {\displaystyle 0=\varnothing }  , 1 = 0 ∪ { 0 } = { ∅ } {\displaystyle 1=0\cup \{0\}=\{\varnothing \}}  , 2 = 1 ∪ { 1 } = { ∅ , { ∅ } } {\displaystyle 2=1\cup \{1\}=\{\varnothing ,\{\varnothing \}\}}  , and so on. The von Neumann construction, along with the axiom of infinity, which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers, N 0 {\displaystyle \mathbb {N} _{0}}  , such that the Peano axioms of arithmetic are satisfied.

In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness follows from the axiom of extensionality. However, the axiom of empty set can be shown redundant in at least two ways:

• Standard first-order logic implies, merely from the logical axioms, that something exists, and in the language of set theory, that thing must be a set. Now the existence of the empty set follows easily from the axiom of separation.
• Even using free logic (which does not logically imply that something exists), there is already an axiom implying the existence of at least one set, namely the axiom of infinity.

Philosophical issues

While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness are debated by philosophers and logicians.

The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something. This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king."[7]

The popular syllogism

is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darling writes that the contrast can be seen by rewriting the statements "Nothing is better than eternal happiness" and "[A] ham sandwich is better than nothing" in a mathematical tone. According to Darling, the former is equivalent to "The set of all things that are better than eternal happiness is ∅ {\displaystyle \varnothing }  " and the latter to "The set {ham sandwich} is better than the set ∅ {\displaystyle \varnothing }  ". The first compares elements of sets, while the second compares the sets themselves.[7]

Jonathan Lowe argues that while the empty set:

it is also the case that:

George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtained by plural quantification over individuals, without reifying sets as singular entities having other entities as members.[9]

• 0 – Number
• Inhabited set – Kind of set in constructive mathematics
• Nothing – Concept denoting the absence of something
• Power set – Mathematical set containing all subsets of a given set

1. ^ a b Weisstein, Eric W. "Empty Set". mathworld.wolfram.com. Retrieved 2020-08-11.
2. ^ "Earliest Uses of Symbols of Set Theory and Logic".
3. ^ Rudin, Walter (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill. p. 300. ISBN 007054235X.
4. ^ "Unicode Standard 5.2" (PDF).
5. ^ e.g. Nina Grønnum (2005, 2013) Fonetik og Fonologi: Almen og dansk. Akademisk forlag, Copenhagen.
6. ^ Bruckner, A.N., Bruckner, J.B., and Thomson, B.S. (2008). Elementary Real Analysis, 2nd edition, p. 9.
7. ^ a b D. J. Darling (2004). The Universal Book of Mathematics. John Wiley and Sons. p. 106. ISBN 0-471-27047-4.
8. ^ E. J. Lowe (2005). Locke. Routledge. p. 87.
9. ^ George Boolos (1984), "To be is to be the value of a variable", The Journal of Philosophy 91: 430–49. Reprinted in 1998, Logic, Logic and Logic (Richard Jeffrey, and Burgess, J., eds.) Harvard University Press, 54–72.

• Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (paperback edition).
• Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN 3-540-44085-2
• Graham, Malcolm (1975), Modern Elementary Mathematics (2nd ed.), Harcourt Brace Jovanovich, ISBN 0155610392

• Weisstein, Eric W. "Empty Set". MathWorld.

The word one can be used as a noun, an adjective, and a pronoun.[2]

It comes from the English word an,[2] which comes from the Proto-Germanic root *ainaz.[2] The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-.[2]

Compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish en, Dutch een, German eins and Old Norse einn.

Compare the Proto-Indo-European root *oi-no- (which means "one, single"[2]) to Greek oinos (which means "ace" on dice[2]), Latin unus (one[2]), Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin and Breton un (one[2]).

One, sometimes referred to as unity,[3][1] is the first non-zero natural number. It is thus the integer after zero.

Any number multiplied by one remains that number, as one is the identity for multiplication. As a result, 1 is its own factorial, its own square and square root, its own cube and cube root, and so on. One is also the result of the empty product, as any number multiplied by one is itself. It is also the only natural number that is neither composite nor prime with respect to division, but is instead considered a unit (meaning of ring theory).

Main article: History of the Hindu–Arabic numeral system

The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Brahmic script of ancient India, where it was a simple vertical line. It was transmitted to Europe via the Maghreb and Andalusia during the Middle Ages, through scholarly works written in Arabic.

In some countries, the serif at the top is sometimes extended into a long upstroke, sometimes as long as the vertical line, which can lead to confusion with the glyph used for seven in other countries. In styles in which the digit 1 is written with a long upstroke, the digit 7 is often written with a horizontal stroke through the vertical line, to disambiguate them. Styles that do not use the long upstroke on digit 1 usually do not use the horizontal stroke through the vertical of the digit 7 either.

While the shape of the character for the digit 1 has an ascender in most modern typefaces, in typefaces with text figures, the glyph usually is of x-height, as, for example, in  .

Many older typewriters lack a separate key for 1, using the lowercase letter l or uppercase I instead. It is possible to find cases when the uppercase J is used, though it may be for decorative purposes. In some typefaces, different glyphs are used for I and 1, but the numeral 1 resembles a small caps version of I, with parallel serifs at top and bottom, with the capital I being full-height.

Mathematically, 1 is:

• in arithmetic (algebra) and calculus, the natural number that follows 0 and the multiplicative identity element of the integers, real numbers and complex numbers;
• more generally, in algebra, the multiplicative identity (also called unity), usually of a group or a ring.

Formalizations of the natural numbers have their own representations of 1. In the Peano axioms, 1 is the successor of 0. In Principia Mathematica, it is defined as the set of all singletons (sets with one element), and in the Von Neumann cardinal assignment of natural numbers, it is defined as the set {0}.

In a multiplicative group or monoid, the identity element is sometimes denoted 1, but e (from the German Einheit, "unity") is also traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are finite fields.

By definition, 1 is the magnitude, absolute value, or norm of a unit complex number, unit vector, and a unit matrix (more usually called an identity matrix). Note that the term unit matrix is sometimes used to mean something quite different.

By definition, 1 is the probability of an event that is absolutely or almost certain to occur.

In category theory, 1 is sometimes used to denote the terminal object of a category.

In number theory, 1 is the value of Legendre's constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting function. Legendre's constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899.

Properties

Tallying is often referred to as "base 1", since only one mark – the tally itself – is needed. This is more formally referred to as a unary numeral system. Unlike base 2 or base 10, this is not a positional notation.

Since the base 1 exponential function (1x) always equals 1, its inverse does not exist (which would be called the logarithm base 1 if it did exist).

There are two ways to write the real number 1 as a recurring decimal: as 1.000..., and as 0.999.... 1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few.

In many mathematical and engineering problems, numeric values are typically normalized to fall within the unit interval from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Likewise, vectors are often normalized into unit vectors (i.e., vectors of magnitude one), because these often have more desirable properties. Functions, too, are often normalized by the condition that they have integral one, maximum value one, or square integral one, depending on the application.

Because of the multiplicative identity, if f(x) is a multiplicative function, then f(1) must be equal to 1.

It is also the first and second number in the Fibonacci sequence (0 being the zeroth) and is the first number in many other mathematical sequences.

The definition of a field requires that 1 must not be equal to 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one element, which is not a singleton and is not a set at all.

1 is the most common leading digit in many sets of data, a consequence of Benford's law.

1 is the only known Tamagawa number for a simply connected algebraic group over a number field.

The generating function that has all coefficients 1 is given by

1 1 − x = 1 + x + x 2 + x 3 + … {\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\ldots }  

This power series converges and has finite value if and only if | x | < 1 {\displaystyle |x|<1}  .

Primality

Main article: Prime number § Primality of one

1 is by convention neither a prime number nor a composite number, but a unit (meaning of ring theory) like −1 and, in the Gaussian integers, i and −i.

The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units. For example, 4 = 22, but if units are included, is also equal to, say, (−1)6 × 123 × 22, among infinitely many similar "factorizations".

1 appears to meet the naïve definition of a prime number, being evenly divisible only by 1 and itself (also 1). As such, some mathematicians considered it a prime number as late as the middle of the 20th century, but mathematical consensus has generally and since then universally been to exclude it for a variety of reasons (such as complicating the fundamental theorem of arithmetic and other theorems related to prime numbers).

1 is the only positive integer divisible by exactly one positive integer, whereas prime numbers are divisible by exactly two positive integers, composite numbers are divisible by more than two positive integers, and zero is divisible by all positive integers.

Table of basic calculations

• The resin identification code used in recycling to identify polyethylene terephthalate.[4]
• The ITU country code for the North American Numbering Plan area, which includes the United States, Canada, and parts of the Caribbean.
• A binary code is a sequence of 1 and 0 that is used in computers for representing any kind of data.
• In many physical devices, 1 represents the value for "on", which means that electricity is flowing.[5][6]
• The numerical value of true in many programming languages.
• 1 is the ASCII code of "Start of Header".

• Dimensionless quantities are also known as quantities of dimension one.
• 1 is the atomic number of hydrogen.
• +1 is the electric charge of positrons and protons.
• Group 1 of the periodic table consists of the alkali metals.
• Period 1 of the periodic table consists of just two elements, hydrogen and helium.
• The dwarf planet Ceres has the minor-planet designation 1 Ceres because it was the first asteroid to be discovered.
• The Roman numeral I often stands for the first-discovered satellite of a planet or minor planet (such as Neptune I, a.k.a. Triton). For some earlier discoveries, the Roman numerals originally reflected the increasing distance from the primary instead.

In the philosophy of Plotinus (and that of other neoplatonists), The One is the ultimate reality and source of all existence.[7] Philo of Alexandria (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers ("De Allegoriis Legum," ii.12 [i.66]).

The Neopythagorean philosopher Nicomachus of Gerasa affirmed that one is not a number, but the source of number. He also believed the number two is the embodiment of the origin of otherness. His number theory was recovered by Boethius in his Latin translation of Nicomachus's treatise Introduction to Arithmetic.[8]

In many professional sports, the number 1 is assigned to the player who is first or leading in some respect, or otherwise important; the number is printed on his sports uniform or equipment. This is the pitcher in baseball, the goalkeeper in association football (soccer), the starting fullback in most of rugby league, the starting loosehead prop in rugby union and the previous year's world champion in Formula One. 1 may be the lowest possible player number, like in the American–Canadian National Hockey League (NHL) since the 1990s[when?] or in American football.

• Number One is Royal Navy informal usage for the chief executive officer of a ship, the captain's deputy responsible for discipline and all normal operation of a ship and its crew.
• 1 is the value of an ace in many playing card games, such as cribbage.
• List of highways numbered 1
• List of public transport routes numbered 1
• 1 is often used to denote the Gregorian calendar month of January.
• 1 CE, the first year of the Common Era
• 01, the former dialling code for Greater London (now 020)
• For Pythagorean numerology (a pseudoscience), the number 1 is the number that means beginning, new beginnings, new cycles, it is a unique and absolute number.
• PRS One, a German paraglider design
• +1 is the code for international telephone calls to countries in the North American Numbering Plan.
• In some countries, a street address of "1" is considered prestigious and developers will attempt to obtain such an address for a building, to the point of lobbying for a street or portion of a street to be renamed, even if this makes the address less useful for wayfinding. The construction of a new street to serve the development may also provide the possibility of a "1" address. An example of such an address is the Apple Campus, located at 1 Infinite Loop, Cupertino, California.

•  Mathematics portal

• −1
• +1 (disambiguation)
• List of mathematical constants
• One (word)
• Root of unity
• List of highways numbered 1

Wikimedia Commons has media related to:
1 (number) (category)

1. ^ a b Weisstein, Eric W. "1". mathworld.wolfram.com. Retrieved 2020-08-10.
2. ^ a b c d e f g h "Online Etymology Dictionary". etymonline.com. Douglas Harper.
3. ^ Skoog, Douglas. Principles of Instrumental Analysis. Brooks/Cole, 2007, p. 758.
4. ^ "Plastic Packaging Resins" (PDF). American Chemistry Council. Archived from the original (PDF) on 2011-07-21.
5. ^ Woodford, Chris (2006), Digital Technology, Evans Brothers, p. 9, ISBN 978-0-237-52725-9
6. ^ Godbole, Achyut S. (1 September 2002), Data Comms & Networks, Tata McGraw-Hill Education, p. 34, ISBN 978-1-259-08223-8
7. ^ Olson, Roger (2017). The Essentials of Christian Thought: Seeing Reality through the Biblical Story. Zondervan Academic. ISBN 9780310521563.
8. ^ British Society for the History of Science (July 1, 1977). "From Abacus to Algorism: Theory and Practice in Medieval Arithmetic". The British Journal for the History of Science. Cambridge University PRess. 10 (2): Abstract. doi:10.1017/S0007087400015375. S2CID 145065082. Retrieved May 16, 2021.

Wikiquote has quotations related to 1 (number).

• The Number 1
• The Positive Integer 1
• Prime curiosities: 1