Empty productFrom Wikipedia, the free encyclopedia

Contents

1 Countable set 11.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Formal definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Empty product 92.1 Nullary arithmetic product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Relevance of defining empty products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.3 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Nullary Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Nullary Cartesian product of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Nullary categorical product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 In logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 In computer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Empty set 133.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

i

ii CONTENTS

3.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Chapter 1

Countable set

“Countable” redirects here. For the linguistic concept, see Count noun.Not to be confused with (recursively) enumerable sets.

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the setof natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, theelements of a countable set can always be counted one at a time and, although the counting may never finish, everyelement of the set is associated with a natural number.Some authors use countable set to mean infinitely countable alone.[1] To avoid this ambiguity, the term at mostcountable may be used when finite sets are included and countably infinite, enumerable, or denumerable[2] oth-erwise.The term countable set was originated by Georg Cantor who contrasted sets which are countable with those which areuncountable (a.k.a. nonenumerable and nondenumerable[3]). Today, countable sets are researched by a branch ofmathematics called discrete mathematics.

1.1 Definition

A set S is called countable if there exists an injective function f from S to the natural numbers N = {0, 1, 2, 3, ...}.[4]

If such an f can be found which is also surjective (and therefore bijective), then S is called countably infinite.In other words, a set is called “countably infinite” if it has one-to-one correspondence with the natural number set, N.As noted above, this terminology is not universal: Some authors use countable to mean what is here called “countablyinfinite,” and to not include finite sets.For alternative (equivalent) formulations of the definition in terms of a bijective function or a surjective function, seethe section Formal definition and properties below.

1.2 History

In the western world, different infinities were first classified by Georg Cantor around 1874.[5]

1.3 Introduction

A set is a collection of elements, and may be described in many ways. One way is simply to list all of its elements;for example, the set consisting of the integers 3, 4, and 5 may be denoted {3, 4, 5}. This is only effective for smallsets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element,sometimes an ellipsis ("...”) is used, if the writer believes that the reader can easily guess what is missing; for example,

1

2 CHAPTER 1. COUNTABLE SET

{1, 2, 3, ..., 100} presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still possibleto list all the elements, because the set is finite.Some sets are infinite; these sets have more than n elements for any integer n. For example, the set of natural numbers,denotable by {0, 1, 2, 3, 4, 5, ...}, has infinitely many elements, and we cannot use any normal number to give itssize. Nonetheless, it turns out that infinite sets do have a well-defined notion of size (or more properly, of cardinality,which is the technical term for the number of elements in a set), and not all infinite sets have the same cardinality.

YX123

x

246

2x

. .

. .

Bijective mapping from integer to even numbers

To understand what this means, we first examine what it does not mean. For example, there are infinitely many oddintegers, infinitely many even integers, and (hence) infinitely many integers overall. However, it turns out that thenumber of even integers, which is the same as the number of odd integers, is also the same as the number of integersoverall. This is because we arrange things such that for every integer, there is a distinct even integer: ... −2→−4,−1→−2, 0→0, 1→2, 2→4, ...; or, more generally, n→2n, see picture. What we have done here is arranged the integersand the even integers into a one-to-one correspondence (or bijection), which is a function that maps between two setssuch that each element of each set corresponds to a single element in the other set.However, not all infinite sets have the same cardinality. For example, Georg Cantor (who introduced this concept)demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.A set is countable if: (1) it is finite, or (2) it has the same cardinality (size) as the set of natural numbers. Equivalently, aset is countable if it has the same cardinality as some subset of the set of natural numbers. Otherwise, it is uncountable.

1.4 Formal definition and properties

By definition a set S is countable if there exists an injective function f : S → N from S to the natural numbers N ={0, 1, 2, 3, ...}.

1.4. FORMAL DEFINITION AND PROPERTIES 3

It might seem natural to divide the sets into different classes: put all the sets containing one element together; all thesets containing two elements together; ...; finally, put together all infinite sets and consider them as having the samesize. This view is not tenable, however, under the natural definition of size.To elaborate this we need the concept of a bijection. Although a “bijection” seems a more advanced concept than anumber, the usual development of mathematics in terms of set theory defines functions before numbers, as they arebased on much simpler sets. This is where the concept of a bijection comes in: define the correspondence

a↔ 1, b↔ 2, c↔ 3

Since every element of {a, b, c} is paired with precisely one element of {1, 2, 3}, and vice versa, this defines abijection.We now generalize this situation and define two sets to be of the same size if (and only if) there is a bijection betweenthem. For all finite sets this gives us the usual definition of “the same size”. What does it tell us about the size ofinfinite sets?Consider the sets A = {1, 2, 3, ... }, the set of positive integers and B = {2, 4, 6, ... }, the set of even positive integers.We claim that, under our definition, these sets have the same size, and that therefore B is countably infinite. Recallthat to prove this we need to exhibit a bijection between them. But this is easy, using n↔ 2n, so that

1 ↔ 2, 2 ↔ 4, 3 ↔ 6, 4 ↔ 8, ....

As in the earlier example, every element of A has been paired off with precisely one element of B, and vice versa.Hence they have the same size. This gives an example of a set which is of the same size as one of its proper subsets,a situation which is impossible for finite sets.Likewise, the set of all ordered pairs of natural numbers is countably infinite, as can be seen by following a path likethe one in the picture:The resulting mapping is like this:

0 ↔ (0,0), 1 ↔ (1,0), 2 ↔ (0,1), 3 ↔ (2,0), 4 ↔ (1,1), 5 ↔ (0,2), 6 ↔ (3,0) ....

It is evident that this mapping will cover all such ordered pairs.Interestingly: if you treat each pair as being the numerator and denominator of a vulgar fraction, then for everypositive fraction, we can come up with a distinct number corresponding to it. This representation includes also thenatural numbers, since every natural number is also a fraction N/1. So we can conclude that there are exactly as manypositive rational numbers as there are positive integers. This is true also for all rational numbers, as can be seen below(a more complex presentation is needed to deal with negative numbers).Theorem: The Cartesian product of finitely many countable sets is countable.This form of triangular mapping recursively generalizes to vectors of finitely many natural numbers by repeatedlymapping the first two elements to a natural number. For example, (0,2,3) maps to (5,3) which maps to 39.Sometimes more than onemapping is useful. This is where youmap the set which you want to show countably infinite,onto another set; and then map this other set to the natural numbers. For example, the positive rational numbers caneasily be mapped to (a subset of) the pairs of natural numbers because p/q maps to (p, q).What about infinite subsets of countably infinite sets? Do these have fewer elements than N?Theorem: Every subset of a countable set is countable. In particular, every infinite subset of a countably infinite setis countably infinite.For example, the set of prime numbers is countable, by mapping the n-th prime number to n:

• 2 maps to 1

• 3 maps to 2

• 5 maps to 3

• 7 maps to 4

4 CHAPTER 1. COUNTABLE SET

1

2

3

01 2 31

2

3

4

5

6

7

8

9

11

12

13 18

17

24

0

0

The Cantor pairing function assigns one natural number to each pair of natural numbers

• 11 maps to 5

• 13 maps to 6

• 17 maps to 7

• 19 maps to 8

• 23 maps to 9

• ...

What about sets being “larger than” N? An obvious place to look would be Q, the set of all rational numbers, whichintuitively may seem much bigger than N. But looks can be deceiving, for we assert:Theorem: Q (the set of all rational numbers) is countable.Q can be defined as the set of all fractions a/b where a and b are integers and b > 0. This can be mapped onto thesubset of ordered triples of natural numbers (a, b, c) such that a ≥ 0, b > 0, a and b are coprime, and c ∈ {0, 1} suchthat c = 0 if a/b ≥ 0 and c = 1 otherwise.

• 0 maps to (0,1,0)

1.4. FORMAL DEFINITION AND PROPERTIES 5

• 1 maps to (1,1,0)

• −1 maps to (1,1,1)

• 1/2 maps to (1,2,0)

• −1/2 maps to (1,2,1)

• 2 maps to (2,1,0)

• −2 maps to (2,1,1)

• 1/3 maps to (1,3,0)

• −1/3 maps to (1,3,1)

• 3 maps to (3,1,0)

• −3 maps to (3,1,1)

• 1/4 maps to (1,4,0)

• −1/4 maps to (1,4,1)

• 2/3 maps to (2,3,0)

• −2/3 maps to (2,3,1)

• 3/2 maps to (3,2,0)

• −3/2 maps to (3,2,1)

• 4 maps to (4,1,0)

• −4 maps to (4,1,1)

• ...

By a similar development, the set of algebraic numbers is countable, and so is the set of definable numbers.Theorem: (Assuming the axiom of countable choice) The union of countably many countable sets is countable.For example, given countable sets a, b, c ...Using a variant of the triangular enumeration we saw above:

• a0 maps to 0

• a1 maps to 1

• b0 maps to 2

• a2 maps to 3

• b1 maps to 4

• c0 maps to 5

• a3 maps to 6

• b2 maps to 7

• c1 maps to 8

• d0 maps to 9

• a4 maps to 10

• ...

6 CHAPTER 1. COUNTABLE SET

Note that this only works if the sets a, b, c,... are disjoint. If not, then the union is even smaller and is therefore alsocountable by a previous theorem.Also note that the axiom of countable choice is needed in order to index all of the sets a, b, c,...Theorem: The set of all finite-length sequences of natural numbers is countable.This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which isa countable set (finite Cartesian product). So we are talking about a countable union of countable sets, which iscountable by the previous theorem.Theorem: The set of all finite subsets of the natural numbers is countable.If you have a finite subset, you can order the elements into a finite sequence. There are only countably many finitesequences, so also there are only countably many finite subsets.The following theorem gives equivalent formulations in terms of a bijective function or a surjective function. A proofof this result can be found in Lang’s text.[2]

Theorem: Let S be a set. The following statements are equivalent:

1. S is countable, i.e. there exists an injective function f : S → N.

2. Either S is empty or there exists a surjective function g : N→ S.

3. Either S is finite or there exists a bijection h : N→ S.

Several standard properties follow easily from this theorem. We present them here tersely. For a gentler presentationsee the sections above. Observe that N in the theorem can be replaced with any countably infinite set. In particularwe have the following Corollary.Corollary: Let S and T be sets.

1. If the function f : S → T is injective and T is countable then S is countable.

2. If the function g : S → T is surjective and S is countable then T is countable.

Proof: For (1) observe that if T is countable there is an injective function h : T → N. Then if f : S → T is injectivethe composition h o f : S → N is injective, so S is countable.For (2) observe that if S is countable there is a surjective function h : N → S. Then if g : S → T is surjective thecomposition g o h : N→ T is surjective, so T is countable.Proposition: Any subset of a countable set is countable.Proof: The restriction of an injective function to a subset of its domain is still injective.Proposition: The Cartesian product of two countable sets A and B is countable.Proof: Note that N × N is countable as a consequence of the definition because the function f : N × N → N givenby f(m, n) = 2m3n is injective. It then follows from the Basic Theorem and the Corollary that the Cartesian productof any two countable sets is countable. This follows because if A and B are countable there are surjections f : N→A and g : N→ B. So

f × g : N × N→ A × B

is a surjection from the countable set N × N to the set A × B and the Corollary implies A × B is countable. This resultgeneralizes to the Cartesian product of any finite collection of countable sets and the proof follows by induction onthe number of sets in the collection.Proposition: The integers Z are countable and the rational numbers Q are countable.Proof: The integers Z are countable because the function f : Z→ N given by f(n) = 2n if n is non-negative and f(n)= 3|n| if n is negative is an injective function. The rational numbers Q are countable because the function g : Z × N→ Q given by g(m, n) = m/(n + 1) is a surjection from the countable set Z × N to the rationals Q.Proposition: If An is a countable set for each n in N then the union of all An is also countable.

1.5. MINIMAL MODEL OF SET THEORY IS COUNTABLE 7

Proof: This is a consequence of the fact that for each n there is a surjective function gn : N → An and hence thefunction

G : N× N →∪n∈N

An

given by G(n, m) = gn(m) is a surjection. Since N × N is countable, the Corollary implies that the union is countable.We are using the axiom of countable choice in this proof in order to pick for each n in N a surjection gn from thenon-empty collection of surjections from N to An.Cantor’s Theorem asserts that if A is a set and P(A) is its power set, i.e. the set of all subsets of A, then there is nosurjective function from A to P(A). A proof is given in the article Cantor’s Theorem. As an immediate consequenceof this and the Basic Theorem above we have:Proposition: The set P(N) is not countable; i.e. it is uncountable.For an elaboration of this result see Cantor’s diagonal argument.The set of real numbers is uncountable (see Cantor’s first uncountability proof), and so is the set of all infinitesequences of natural numbers. A topological proof for the uncountability of the real numbers is described at finiteintersection property.

1.5 Minimal model of set theory is countable

If there is a set that is a standard model (see inner model) of ZFC set theory, then there is a minimal standardmodel (see Constructible universe). The Löwenheim-Skolem theorem can be used to show that this minimal modelis countable. The fact that the notion of “uncountability” makes sense even in this model, and in particular that thismodel M contains elements which are

• subsets of M, hence countable,

• but uncountable from the point of view of M,

was seen as paradoxical in the early days of set theory, see Skolem’s paradox.The minimal standard model includes all the algebraic numbers and all effectively computable transcendental num-bers, as well as many other kinds of numbers.

1.6 Total orders

Countable sets can be totally ordered in various ways, e.g.:

• Well orders (see also ordinal number):

• The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...)• The integers in the order (0, 1, 2, 3, ...; −1, −2, −3, ...)

• Other (not well orders):

• The usual order of integers (..., −3, −2, −1, 0, 1, 2, 3, ...)• The usual order of rational numbers (Cannot be explicitly written as a list!)

Note that in both examples of well orders here, any subset has a least element; and in both examples of non-wellorders, some subsets do not have a least element. This is the key definition that determines whether a total order isalso a well order.

8 CHAPTER 1. COUNTABLE SET

1.7 See also• Aleph number

• Counting

• Hilbert’s paradox of the Grand Hotel

• Uncountable set

1.8 Notes[1] For an example of this usage see (Rudin 1976, Chapter 2).

[2] See (Lang 1993, §2 of Chapter I).

[3] See (Apostol 1969, Chapter 13.19).

[4] Since there is an obvious bijection between N and N* = {1, 2, 3, ...}, it makes no difference whether one considers 0 tobe a natural number or not. In any case, this article follows ISO 31-11 and the standard convention in mathematical logic,which make 0 a natural number.

[5] Stillwell, John C. (2010), Roads to Infinity: TheMathematics of Truth and Proof, CRC Press, p. 10, ISBN 9781439865507,Cantor’s discovery of uncountable sets in 1874 was one of the most unexpected events in the history of mathematics. Before1874, infinity was not even considered a legitimate mathematical subject by most people, so the need to distinguish betweencountable and uncountable infinities could not have been imagined.

1.9 References• Lang, Serge (1993), Real and Functional Analysis, Berlin, New York: Springer-Verlag, ISBN 0-387-94001-4

• Rudin, Walter (1976), Principles of Mathematical Analysis, New York: McGraw-Hill, ISBN 0-07-054235-X

• Apostol, Tom M. (June 1969),Multi-Variable Calculus and Linear Algebra with Applications, Calculus 2 (2nded.), New York: John Wiley + Sons, ISBN 978-0-471-00007-5

1.10 External links• Weisstein, Eric W., “Countable Set”, MathWorld.

Chapter 2

Empty product

In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is by conventionequal to the multiplicative identity 1 (assuming there is an identity for the multiplication operation in question), justas the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.[1][2][3]

The term “empty product” is most often used in the above sense when discussing arithmetic operations. However,the term is sometimes employed when discussing set-theoretic intersections, categorical products, and products incomputer programming; these are discussed below.

2.1 Nullary arithmetic product

2.1.1 Justification

Let a1, a2, a3,... be a sequence of numbers, and let

Pm =

m∏i=1

ai = a1 · · · am

be the product of the first m elements of the sequence. Then

Pm = am · Pm−1

for all m = 1,2,... provided that we use the following conventions: P1 = a1 and P0 = 1 . In other words, a “product”P1 with only one factor evaluates to that factor, while a “product” P0 with no factors at all evaluates to 1. Allowing a“product” with only one or zero factors reduces the number of cases to be considered in many mathematical formulas.Such “products” are natural starting points in induction proofs, as well as in algorithms. For these reasons, the “emptyproduct is one convention” is common practice in mathematics and computer programming.

2.1.2 Relevance of defining empty products

The notion of an empty product is useful for the same reason that the number zero and the empty set are useful: whilethey seem to represent quite uninteresting notions, their existence allows for a much shorter mathematical presentationof many subjects.For example, the empty products 0! = 1 and x0 = 1 shorten Taylor series notation (see zero to the power of zero for adiscussion when x=0). Likewise, if M is an n × n matrix then M0 is the n × n identity matrix, reflecting the fact thatapplying a linear map zero times has the same effect as applying the identity map.As another example, the fundamental theorem of arithmetic says that every positive integer can be written uniquelyas a product of primes. However, if we do not allow products with only 0 or 1 factors, then the theorem (and itsproof!) become longer.[4][5]

9

10 CHAPTER 2. EMPTY PRODUCT

More examples of the use of the empty product in mathematics may be found in the binomial theorem (which assumesand implies that x0=1 for all x), Stirling number, König’s theorem, binomial type, binomial series, difference operatorand Pochhammer symbol.

2.1.3 Logarithms

Since logarithms turn products into sums, they should map an empty product to an empty sum. So if we define theempty product to be 1, then the empty sum should be ln(1) = 0 . Conversely, the exponential function turns sumsinto products, so if we define the empty sum to be 0, then the empty product should be e0 = 1 .

∏i

xi = e∑

i ln xi

2.2 Nullary Cartesian product

Consider the general definition of the Cartesian product:

∏i∈I

Xi = {g : I →∪i∈I

Xi | ∀i g(i) ∈ Xi}.

If I is empty, the only such g is the empty function f∅ , which is the unique subset of∅×∅ that is a function∅ → ∅, namely the empty subset ∅ (the only subset that ∅×∅ = ∅ has):

∏∅

= {f∅ : ∅ → ∅} = {∅}.

Thus, the cardinality of the Cartesian product of no sets is 1.Under the perhaps more familiar n-tuple interpretation,

∏∅

= {()},

that is, the singleton set containing the empty tuple. Note that in both representations the empty product has cardinality1.

2.2.1 Nullary Cartesian product of functions

The empty Cartesian product of functions is again the empty function.

2.3 Nullary categorical product

In any category, the product of an empty family is a terminal object of that category. This can be demonstrated byusing the limit definition of the product. An n-fold categorical product can be defined as the limit with respect to adiagram given by the discrete category with n objects. An empty product is then given by the limit with respect tothe empty category, which is the terminal object of the category if it exists. This definition specializes to give resultsas above. For example, in the category of sets the categorical product is the usual Cartesian product, and the terminalobject is a singleton set. In the category of groups the categorical product is the Cartesian product of groups, and theterminal object is a trivial group with one element. To obtain the usual arithmetic definition of the empty product wemust take the decategorification of the empty product in the category of finite sets.Dually, the coproduct of an empty family is an initial object. Nullary categorical products or coproducts may notexist in a given category; e.g. in the category of fields, neither exists.

2.4. IN LOGIC 11

2.4 In logic

Classical logic defines the operation of conjunction, which is generalized to universal quantification in and predicatecalculus, and is widely known as logical multiplication because we intuitively identify true with 1 and false with 0 andour conjunction behaves as ordinary multiplier. Multipliers can have arbitrary number of inputs. In case of 0 inputs,we have empty conjunction, which is identically equal to true.This is related to another concept in logic, vacuous truth, which tells us that empty set of objects can have anyproperty. It can be explained the way that the conjunction (as part of logic in general) deals with values less or equal1. This means that longer is the conjunction, the higher is probability to end up with 0. Conjunction merely checksthe propositions and returns 0 (or false) as soon as one of propositions evaluates to false. Reducing the number ofconjoined propositions increases the chance to pass the check and stay with 1. Particularly, if there are 0 tests ormembers to check, none can fail so, by default, we must always succeed regardless of which propositions or memberproperties had to be tested.

2.5 In computer programming

Many programming languages, such as Python, allow the direct expression of lists of numbers, and even functionsthat allow an arbitrary number of parameters. If such a language has a function that returns the product of all thenumbers in a list, it usually works like this:listprod( [2,3,5] ) --> 30 listprod( [2,3] ) --> 6 listprod( [2] ) --> 2 listprod( [] ) --> 1This convention helps avoid having to code special cases like “if length of list is 1” or “if length of list is zero” asspecial cases.Multiplication is an infix operator and therefore a binary operator, complicating the notation of an empty product.Some programming languages handle this by implementing variadic functions. For example, the fully parenthesizedprefix notation of Lisp languages gives rise to a natural notation for nullary functions:(* 2 2 2) ; evaluates to 8 (* 2 2) ; evaluates to 4 (* 2) ; evaluates to 2 (*) ; evaluates to 1

2.6 See also

• Iterated binary operation

• Empty sum

2.7 References

[1] Jaroslav Nešetřil, Jiří Matoušek (1998). Invitation to Discrete Mathematics. Oxford University Press. p. 12. ISBN 0-19-850207-9.

[2] A.E. Ingham and R C Vaughan (1990). The Distribution of Prime Numbers. Cambridge University Press. p. 1. ISBN0-521-39789-8.

[3] Page 9 of Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556

[4] EdsgerWybe Dijkstra (1990-03-04). “How Computing Science created a newmathematical style”. EWD. Retrieved 2010-01-20. Hardy and Wright: “Every positive integer, except 1, is a product of primes”, Harold M. Stark: “If n is an integergreater than 1, then either n is prime or n is a finite product of primes.”. These examples —which I owe to A.J.M. vanGasteren— both reject the empty product, the last one also rejects the product with a single factor.

[5] Edsger Wybe Dijkstra (1986-11-14). “The nature of my research and why I do it”. EWD. Retrieved 2010-07-03. Butalso 0 is certainly finite and by defining the product of 0 factors —how else?— to be equal to 1 we can do away with theexception: “If n is a positive integer, then n is a finite product of primes.”

12 CHAPTER 2. EMPTY PRODUCT

2.8 External links• PlanetMath article on the empty product

Chapter 3

Empty set

"∅" redirects here. For similar symbols, see Ø (disambiguation).In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size orcardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists byincluding an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of setsare trivially true for the empty set.Null set was once a common synonym for “empty set”, but is now a technical term in measure theory. The empty setmay also be called the void set.

3.1 Notation

Common notations for the empty set include "{}", "∅", and " ∅ ". The latter two symbols were introduced by theBourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Norwegian and Danish alphabets(and not related in any way to the Greek letter Φ).[1]

The empty-set symbol ∅ is found at Unicode point U+2205.[2] In TeX, it is coded as \emptyset or \varnothing.

3.2 Properties

In standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements;therefore there can be only one set with no elements. Hence there is but one empty set, and we speak of “the emptyset” rather than “an empty set”.The mathematical symbols employed below are explained here.For any set A:

• The empty set is a subset of A:

∀A : ∅ ⊆ A

• The union of A with the empty set is A:

∀A : A ∪ ∅ = A

• The intersection of A with the empty set is the empty set:

∀A : A ∩ ∅ = ∅

• The Cartesian product of A and the empty set is the empty set:

∀A : A× ∅ = ∅

13

14 CHAPTER 3. EMPTY SET

The empty set is the set containing no elements.

The empty set has the following properties:

• Its only subset is the empty set itself:

∀A : A ⊆ ∅ ⇒ A = ∅

• The power set of the empty set is the set containing only the empty set:

2∅ = {∅}

3.2. PROPERTIES 15

A symbol for the empty set

• Its number of elements (that is, its cardinality) is zero:

card(∅) = 0

The connection between the empty set and zero goes further, however: in the standard set-theoretic definition ofnatural numbers, we use sets to model the natural numbers. In this context, zero is modelled by the empty set.For any property:

• For every element of ∅ the property holds (vacuous truth);

• There is no element of ∅ for which the property holds.

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

• For every element of V the property holds;

• There is no element of V for which the property holds,

V = ∅

16 CHAPTER 3. EMPTY SET

By the definition of subset, the empty set is a subset of any set A, as every element x of ∅ belongs to A. If it is nottrue that every element of ∅ is in A, there must be at least one element of ∅ that is not present in A. Since there areno elements of ∅ at all, there is no element of ∅ that is not in A. Hence every element of ∅ is in A, and ∅ is a subsetof A. Any statement that begins “for every element of ∅ " 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.”

3.2.1 Operations on the empty set

Operations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the sumof the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product).Ultimately, the results of these operations say more about the operation in question than about the empty set. Forinstance, zero is the identity element for addition, and one is the identity element for multiplication.A disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is adisarrangment of itself as no element can be found that retains its original position.

3.3 In other areas of mathematics

3.3.1 Extended real numbers

Since the empty set has no members, when it is considered as a subset of any ordered set, then every member ofthat set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of thereal numbers, with its usual ordering, represented by the real number line, every real number is both an upper andlower bound for the empty set.[3] When considered as a subset of the extended reals formed by adding two “numbers”or “points” to the real numbers, namely negative infinity, denoted −∞, which is defined to be less than every otherextended real number, and positive infinity, denoted +∞, which is defined to be greater than every other extendedreal number, then:

sup ∅ = min({−∞,+∞} ∪ R) = −∞,

and

inf ∅ = max({−∞,+∞} ∪ R) = +∞.

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 infinityis the identity element for the maximum and supremum operators, while positive infinity is the identity element forminimum and infimum.

3.3.2 Topology

Considered as a subset of the real number line (or more generally any topological space), the empty set is both closedand open; it is an example of a “clopen” set. All its boundary points (of which there are none) are in the empty set,and the set is therefore closed; while for every one of its points (of which there are again none), there is an openneighbourhood in the empty set, and the set is therefore open. Moreover, the empty set is a compact set by the factthat every finite set is compact.The closure of the empty set is empty. This is known as “preservation of nullary unions.”

3.3.3 Category theory

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

3.4. QUESTIONED EXISTENCE 17

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

3.4 Questioned existence

3.4.1 Axiomatic set theory

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

• There is already an axiom implying the existence of at least one set. Given such an axiom together with theaxiom of separation, the existence of the empty set is easily proved.

• In the presence of urelements, it is easy to prove that at least one set exists, viz. the set of all urelements. Again,given the axiom of separation, the empty set is easily proved.

3.4.2 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) explainsthat the empty set is not nothing, but rather “the set of all triangles with four sides, the set of all numbers that arebigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king.”[4]

The popular syllogism

Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sand-wich is better than eternal happiness

is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darlingwrites 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 “Theset of all things that are better than eternal happiness is ∅ " and the latter to “The set {ham sandwich} is better thanthe set ∅ ". It is noted that the first compares elements of sets, while the second compares the sets themselves.[4]

Jonathan Lowe argues that while the empty set:

"...was undoubtedly an important landmark in the history of mathematics, … we should not assume thatits utility in calculation is dependent upon its actually denoting some object.”

it is also the case that:

“All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and(3) is unique amongst sets in having no members. However, there are very many things that 'have nomembers’, in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things haveno members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a setwhich has no members. We cannot conjure such an entity into existence by mere stipulation.”[5]

George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtainedby plural quantification over individuals, without reifying sets as singular entities having other entities as members.[6]

18 CHAPTER 3. EMPTY SET

3.5 See also• Inhabited set

• Nothing

3.6 Notes[1] Earliest Uses of Symbols of Set Theory and Logic.

[2] Unicode Standard 5.2

[3] Bruckner, A.N., Bruckner, J.B., and Thomson, B.S., 2008. Elementary Real Analysis, 2nd ed. Prentice Hall. P. 9.

[4] D. J. Darling (2004). The universal book of mathematics. John Wiley and Sons. p. 106. ISBN 0-471-27047-4.

[5] E. J. Lowe (2005). Locke. Routledge. p. 87.

[6] • George Boolos, 1984, “To be is to be the value of a variable,” The Journal of Philosophy 91: 430–49. Reprinted inhis 1998 Logic, Logic and Logic (Richard Jeffrey, and Burgess, J., eds.) Harvard Univ. Press: 54–72.

3.7 References• 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, ISBN3-540-44085-2

• Graham, Malcolm (1975), Modern Elementary Mathematics (HARDCOVER) (in English) (2nd ed.), NewYork: Harcourt Brace Jovanovich, ISBN 0155610392

3.8 External links• Weisstein, Eric W., “Empty Set”, MathWorld.

3.9. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 19

3.9 Text and image sources, contributors, and licenses

3.9.1 Text• Countable set Source: https://en.wikipedia.org/wiki/Countable_set?oldid=665835456 Contributors: Damian Yerrick, AxelBoldt, Bryan

Derksen, Zundark, Xaonon, Danny, Oliverkroll, Kurt Jansson, Stevertigo, Patrick, Michael Hardy, Kku, Александър, Revolver, CharlesMatthews, Dysprosia, Hyacinth, Fibonacci, Head, Aleph4, Robbot, Romanm, MathMartin, Paul Murray, Ruakh, Tobias Bergemann,Giftlite, Everyking, Georgesawyer, Wyss, Simon Lacoste-Julien, Jorend, Hkpawn~enwiki, TheObtuseAngleOfDoom, Possession, Noisy,Rich Farmbrough, Pak21, Paul August, Gauge, Aranel, El C, PhilHibbs, Robotje, Jumbuck, Keenan Pepper, ABCD, Hu, Julioc, LOL,MattGiuca, Esben~enwiki, Graham87, Jetekus, Josh Parris, Salix alba, VKokielov, Kri, Chobot, Roboto de Ajvol, YurikBot, Taejo,Trovatore, Muu-karhu, Scs, Hirak 99, Lt-wiki-bot, Arthur Rubin, Reyk, Benandorsqueaks, Teply, Brentt, SmackBot, David Shear,Brick Thrower, Canthusus, Edonovan, Edgar181, Grokmoo, Xie Xiaolei, Silly rabbit, SEIBasaurus, Octahedron80, Javalenok, NYKevin,Matchups, SundarBot, Grover cleveland, Richard001, Bidabadi~enwiki, SashatoBot, Cronholm144, The Infidel, 16@r, Mets501, Newone,S0me l0ser, Martin Kozák, JRSpriggs, CRGreathouse, CBM, Mct mht, Gregbard, FilipeS, Thijs!bot, Colin Rowat, Magioladitis, Usien6,David Eppstein, MartinBot, Wdevauld, R'n'B, Ttwo, Qatter, Owlgorithm, Stokkink, Alyssa kat13, LokiClock, Rei-bot, Anonymous Dis-sident, Pieman93, Ilyaroz, Mike4ty4, SieBot, Caltas, Krishna.91, JackSchmidt, Unitvoice, Sunrise, Ken123BOT, Gigacephalus, Drag-onBot, Alexbot, Hans Adler, HumphreyW, Addbot, Topology Expert, Tomthecool, LaaknorBot, Super duper jimbo, LinkFA-Bot, Jarble,JakobVoss, Legobot, Luckas-bot, Yobot, AnomieBOT, Mihnea Maftei, Materialscientist, ArthurBot, Bdmy, Omnipaedista, Johnfranks,Worldrimroamer, Tkuvho, Phanxan, RedBot, Raiden09, EmausBot, QuantumOfHistory, Vishwaraj.anand00, Imcrazyaboutyou, AccessDenied, ClueBot NG, Wcherowi, Misshamid, Widr, Helpful Pixie Bot, Sinestar, Jim Sukwutput, Adityapanwarr, Dexbot, Sriharsh1234,Jochen Burghardt, Namespan, OliverBel, Matthew Kastor, Niceguy6, HKennethB and Anonymous: 127

• Empty product Source: https://en.wikipedia.org/wiki/Empty_product?oldid=655829085 Contributors: Damian Yerrick, Toby Bartels,Fubar Obfusco, Patrick, Michael Hardy, Komap, Paul A, Eric119, Ams80, Cyan, Revolver, Charles Matthews, WhisperToMe, McKay,Phil Boswell, Robbot, Fredrik, Altenmann, Henrygb, Wile E. Heresiarch, Giftlite, Paisa, ShaunMacPherson, Fropuff, Sundar, Cam-byses, Eequor, Fak119, Matt Crypto, CryptoDerk, Rlcantwell, Smyth, Paul August, Susvolans, Grick, Army1987, C S, La goutte depluie, Shreevatsa, Uncle G, Apokrif, MFH, Marudubshinki, Qwertyus, Jshadias, Chenxlee, Bubba73, Moskvax, Mathbot, Flashmor-bid, Trovatore, Nishantman, Ms2ger, WAS 4.250, Reyk, Bo Jacoby, SmackBot, InverseHypercube, Melchoir, Eskimbot, NoJoy, Oc-tahedron80, Javalenok, NYKevin, Daniel-Dane, Leland McInnes, Cybercobra, Daqu, MvH, EdC~enwiki, Happy-melon, Maxcantor,JRSpriggs, CBM, HenningThielemann, Cydebot, Headbomb, Dfrg.msc, RobHar, Ricardo sandoval, CommonsDelinker, Daniel5Ko, Os-sido, Steel1943, TXiKiBoT, Tom239, Anonymous Dissident, Dmcq, Thehotelambush, ClueBot, Watchduck, ChrisHodgesUK, Addbot,Ozob, Xario, ב ,.דניאל PV=nRT, Yobot, Citation bot, Charvest, D'ohBot, Citation bot 1, 777sms, Ebehn, Helpful Pixie Bot, Macofe andAnonymous: 43

• Empty set Source: https://en.wikipedia.org/wiki/Empty_set?oldid=666003197 Contributors: AxelBoldt, Lee Daniel Crocker, Uriyan,Bryan Derksen, Tarquin, Jeronimo, Andre Engels, XJaM, Christian List, Toby~enwiki, Toby Bartels, Ryguasu, Hephaestos, Patrick,Michael Hardy,MartinHarper, TakuyaMurata, Eric119, Den fjättrade ankan~enwiki, Andres, Evercat, Renamed user 4, CharlesMatthews,Berteun, Dcoetzee, David Latapie, Dysprosia, Jitse Niesen, Krithin, Hyacinth, Spikey, Jeanmichel~enwiki, Flockmeal, Phil Boswell,Robbot, Sanders muc, Peak, Romanm, Gandalf61, Henrygb, Wikibot, Pengo, Tobias Bergemann, Adam78, Tosha, Giftlite, Dbenbenn,Vfp15, BenFrantzDale, Herbee, Fropuff, MichaelHaeckel, Macrakis, Python eggs, Rdsmith4, Mike Rosoft, Brianjd, Mormegil, Guan-abot, Paul August, Spearhead, EmilJ, BrokenSegue, Nortexoid, 3mta3, Obradovic Goran, Jonathunder, ABCD, Sligocki, Dzhim, Itsmine,HenryLi, Hq3473, Angr, Isnow, Qwertyus, MarSch, Salix alba, Bubba73, ChongDae, Salvatore Ingala, Chobot, YurikBot, RussBot,Rsrikanth05, Trovatore, Ms2ger, Saric, EtherealPurple, GrinBot~enwiki, TomMorris, SmackBot, InverseHypercube, Melchoir, FlashSh-eridan, Ohnoitsjamie, Joefaust, SMP, J. Spencer, Octahedron80, Iit bpd1962, Tamfang, Cybercobra, Dreadstar, RandomP, Jon Awbrey,Jóna Þórunn, Lambiam, Jim.belk, Vanished user v8n3489h3tkjnsdkq30u3f, Loadmaster, Hvn0413, Mets501, EdC~enwiki, Joseph Solisin Australia, Spindled, James pic, Amalas, Philiprbrenan, CBM, Gregbard, Cydebot, Pais, Julian Mendez, Malleus Fatuorum, Epbr123,Nick Number, Escarbot, Sluzzelin, .anacondabot, David Eppstein, Ttwo, Maurice Carbonaro, Ian.thomson, It Is Me Here, Daniel5Ko,NewEnglandYankee, DavidCBryant, VolkovBot, Zanardm, Rei-bot, Anonymous Dissident, Andy Dingley, SieBot, Niv.sarig, ToePeu.bot,Randomblue, Niceguyedc, Wounder, Nosolution182, Versus22, Palnot, AmeliaElizabeth, Feinoha, American Eagle, ThisIsMyWikipedi-aName, LaaknorBot, AnnaFrance, Numbo3-bot, Zorrobot, Legobot, Luckas-bot, Yobot, Ciphers, Xqbot, Nasnema, , GrouchoBot,LucienBOT, Pinethicket, Kiefer.Wolfowitz, Abductive, Jauhienij, FoxBot, Lotje, LilyKitty, Woodsy dong peep, EmausBot, Sharlack-Hames, Ystory, ClueBot NG, Cntras, Rezabot, Helpful Pixie Bot, Michael.croghan, Langing, Ugncreative Usergname, JYBot, Kephir,Phinumu, Noyster, GeoffreyT2000, Skw27 and Anonymous: 82

3.9.2 Images• File:Aplicación_2_inyectiva_sobreyectiva02.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/69/Aplicaci%C3%B3n_

2_inyectiva_sobreyectiva02.svg License: Public domain Contributors: Own work Original artist: HiTe

• File:Empty_set.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/aa/Empty_set.svg License: Public domain Contribu-tors: Own work Original artist: Octahedron80

• File:Nullset.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6a/Nullset.svg License: CC BY-SA 3.0 Contributors: Ownwork Original artist: Hugo Férée

• File:Pairing_natural.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6f/Pairing_natural.svg License: CC-BY-SA-3.0Contributors: Own work Original artist: Cronholm144

• File:Venn_A_intersect_B.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6d/Venn_A_intersect_B.svg License: Pub-lic domain Contributors: Own work Original artist: Cepheus

• File:Wiktionary-logo-en.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f8/Wiktionary-logo-en.svg License: Publicdomain Contributors: Vector version of Image:Wiktionary-logo-en.png. Original artist: Vectorized by Fvasconcellos (talk · contribs),based on original logo tossed together by Brion Vibber

20 CHAPTER 3. EMPTY SET

3.9.3 Content license• Creative Commons Attribution-Share Alike 3.0