Notes on the combinatorial fundamentals of algebra

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fundamentals of algebra ^∗

Darij Grinberg May 25, 2021

(with minor corrections May 25, 2021)

^†

Abstract. This is a detailed survey – with rigorous and self-contained proofs – of some of the basics of elementary combinatorics and algebra, including the properties of finite sums, binomial coefficients, permutations and determinants. It is entirely expository (and written to a large extent as a repository for folklore proofs); no new results (and few, if any, new proofs) appear.

1. Introduction

These notes are a detailed introduction to some of the basic objects of combinatorics and algebra: finite sums, binomial coefficients, permutations and determinants (from a combinatorial viewpoint – no linear algebra is presumed). To a lesser extent, modular arithmetic and recurrent integer sequences are treated as well.

The reader is assumed to be proficient in high-school mathematics, and mature enough to understand nontrivial mathematical proofs. Familiarity with “contest mathematics” is also useful.

One feature of these notes is their focus on rigorous and detailed proofs. In- deed, so extensive are the details that a reader with experience in mathematics will probably be able to skip whole paragraphs of proof without losing the thread.

(As a consequence of this amount of detail, the notes contain far less material than might be expected from their length.) Rigorous proofs mean that (with some minor exceptions) no “handwaving” is used; all relevant objects are defined in mathematical (usually set-theoretical) language, and are manipulated in logically well-defined ways. (In particular, some things that are commonly taken for granted in the liter- ature – e.g., the fact that the sum of n numbers is well-defined without specifying in what order they are being added – are unpacked and proven in a rigorous way.)

These notes are split into several chapters:

• Chapter 1 collects some basic facts and notations that are used in later chapters. This chapter is not meant to be read first; it is best consulted when needed.

• Chapter 2 is an in-depth look at mathematical induction (in various forms, including strong and two-sided induction) and several of its applications (including basic modular arithmetic, division with remainder, Bezout’s theorem, some properties of recurrent sequences, the well-definedness of compositions ofnmaps and sums of nnumbers, and various properties thereof).

• Chapter 3 surveys binomial coefficients and their basic properties. Unlike most texts on combinatorics, our treatment of binomial coefficients leans to the algebraic side, relying mostly on computation and manipulations of sums;

but some basics of counting are included.

• Chapter 4 treats some more properties of Fibonacci-like sequences, including explicit formulas (à la Binet) for two-term recursions of the formxn =axn−1+ bxn−2.

• Chapter 5 is concerned with permutations of finite sets. The coverage is heav- ily influenced by the needs of the next chapter (on determinants); thus, a great role is played by transpositions and the inversions of a permutation.

• Chapter 6 is a comprehensive introduction to determinants of square matrices

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over a commutative ring¹, from an elementary point of view. This is probably the most unique feature of these notes: I define determinants using Leib- niz’s formula (i.e., as sums over permutations) and prove all their properties (Laplace expansion in one or several rows; the Cauchy-Binet, Desnanot-Jacobi and Plücker identities; the Vandermonde and Cauchy determinants; and several more) from this vantage point, thus treating them as an elementary object unmoored from its linear-algebraic origins and applications. No use is made of modules (or vector spaces), exterior powers, eigenvalues, or of the

“universal coefficients” trick². (This means that all proofs are done through combinatorics and manipulation of sums – a rather restrictive requirement!) This is a conscious and (to a large extent) aesthetic choice on my part, and I donot consider it the best way to learn about determinants; but I do regard it as a road worth charting, and these notes are my attempt at doing so.

The notes include numerous exercises of varying difficulty, many of them solved.

The reader should treat exercises and theorems (and propositions, lemmas and corollaries) as interchangeable to some extent; it is perfectly reasonable to read the solution of an exercise, or conversely, to prove a theorem on one’s own instead of reading its proof. The reader’s experience will be the strongest determinant of their success in solving the exercises independently.

I have not meant these notes to be a textbook on any particular subject. For one thing, their content does not map to any of the standard university courses, but rather straddles various subjects:

• Much of Chapter 3 (on binomial coefficients) and Chapter 5 (on permutations) is seen in a typical combinatorics class; but my focus is more on the algebraic side and not so much on the combinatorics.

• Chapter 6 studies determinants far beyond what a usual class on linear algebra would do; but it does not include any of the other topics that a linear algebra class usually covers (such as row reduction, vector spaces, linear maps, eigenvectors, tensors or bilinear forms).

• Being devoted to mathematical induction, Chapter 2 appears to cover the same ground as a typical “introduction to proofs” textbook or class (or at least one of its main topics). In reality, however, it complements rather than competes with most “introduction to proofs” texts I have seen; the examples I give are (with a few exceptions) nonstandard, and the focus different.

1The notion of a commutative ring is defined (and illustrated with several examples) in Section 6.1, but I don’t delve deeper into abstract algebra.

2This refers to the standard trick used for proving determinant identities (and other polynomial identities), in which one first replaces the entries of a matrix (or, more generally, the variables appearing in the identity) by indeterminates, then uses the “genericity” of these indeterminates (e.g., to invert the matrix, or to divide by an expression that could otherwise be 0), and finally substitutes the old variables back for the indeterminates.

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• While the notions of rings and groups are defined in Chapter 6, I cannot claim to really be doing any abstract algebra: I am merely working in rings (i.e., doing computations with elements of rings or with matrices over rings), rather than working with rings. Nevertheless, Chapter 6 might help familiarize the reader with these concepts, facilitating proper learning of abstract algebra later on.

All in all, these notes are probably more useful as a repository of detailed proofs than as a textbook to be read cover-to-cover. Indeed, one of my motives in writing them was to have a reference for certain folklore results – one in which these results are proved elementary and without appeal to the reader’s problem-solving acumen.

These notes began as worksheets for the PRIMES reading project I have mentored in 2015; they have since been greatly expanded with new material (some of it origi- nally written for my combinatorics classes, some in response to math.stackexchange questions).

The notes are in flux, and probably have their share of misprints. I thank Anya Zhang and Karthik Karnik (the two students taking part in the 2015 PRIMES project) for finding some errors. Thanks also to the PRIMES project at MIT, which gave the impetus for the writing of this notes; and to George Lusztig for the spon- sorship of my mentoring position in this project.

1.1. Prerequisites

Let me first discuss the prerequisites for a reader of these notes. At the current moment, I assume that the reader

• has a good grasp on basic school-level mathematics (integers, rational numbers, etc.);

• has some experience with proofs (mathematical induction, proof by contra- diction, the concept of “WLOG”, etc.) and mathematical notation (functions, subscripts, cases, what it means for an object to be “well-defined”, etc.)³;

3A great introduction into these matters (and many others!) is the free book [LeLeMe16] by Lehman, Leighton and Meyer. (Practical note:As of 2018, this book is still undergoing frequent revisions; thus, the version I am citing below might be outdated by the time you are reading this. I therefore suggest searching for possibly newer versions on the internet. Unfortunately, you will also find many older versions, often as the first google hits. Try searching for the title of the book along with the current year to find something up-to-date.)

Another introduction to proofs and mathematical workmanship is Day’s [Day16] (but beware that the definition of polynomials in [Day16, Chapter 5] is the wrong one for our purposes).

Two others are Hammack’s [Hammac15] and Doud’s and Nielsen’s [DouNie19]. Yet another is Newstead’s [Newste19] (currently a work in progress, but promising to become one of the most interesting and sophisticated texts of this kind). There are also several books on this subject; an especially popular one is Velleman’s [Vellem06].

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• knows what a polynomial is (at least over Z and Q) and how polynomials differ from polynomial functions⁴;

• is somewhat familiar with the summation sign (∑) and the product sign (∏) and knows how to transform them (e.g., interchanging summations, and sub- stituting the index)⁵;

• has some familiarity with matrices (i.e., knows how to add and to multiply them)⁶.

Probably a few more requirements creep in at certain points of the notes, which I have overlooked. Some examples and remarks rely on additional knowledge (such as analysis, graph theory, abstract algebra); however, these can be skipped.

1.2. Notations

• In the following, we use N to denote the set {0, 1, 2, . . .}. (Be warned that some other authors use the letterNfor{1, 2, 3, . . .} instead.)

• We letQdenote the set of all rational numbers; we letRbe the set of all real numbers; we letCbe the set of all complex numbers⁷.

• If X and Y are two sets, then we shall use the notation “X → Y, x 7→ E”

(wherexis some symbol which has no specific meaning in the current context, and whereEis some expression which usually involves x) for “the map from X toYwhich sends every x∈ X toE”.

For example, “N→_N, x 7→ x²+x+6” means the map from NtoNwhich sends every x∈ _N_tox²+x+_6.

For another example, “N → _Q, x 7→ ^x

1+x” denotes the map from Nto Q which sends every x∈ _N_to ^x

1+x. ⁸

4This is used only in a few sections and exercises, so it is not an unalienable requirement. See Section 1.5 below for a quick survey of polynomials, and for references to sources in which precise definitions can be found.

5See Section 1.4 below for a quick overview of the notations that we will need.

6See, e.g., [Grinbe16b, Chapter 2] or any textbook on linear algebra for an introduction.

7See [Swanso20, Section 3.9] or [AmaEsc05, Section I.11] for a quick introduction to complex numbers. We will rarely use complex numbers. Most of the time we use them, you can instead use real numbers.

8A word of warning: Of course, the notation “X → Y, x 7→ E” does not always make sense;

indeed, the map that it stands for might sometimes not exist. For instance, the notation “N→ Q, x 7→ ^x

1−x” does not actually define a map, because the map that it is supposed to define (i.e., the map fromNtoQwhich sends everyx∈_Nto x

1−x) does not exist (since x

1−x is not defined for x = 1). For another example, the notation “N →Z, x 7→ ^x

1+x” does not define

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• If S is a set, then the powerset of S means the set of all subsets of S. This powerset will be denoted by P(S). For example, the powerset of {1, 2} is P({1, 2}) ={∅,{1},{2},{1, 2}}.

• The letter i willnotdenote the imaginary unit√

−1 (except when we explicitly say so).

Further notations will be defined whenever they arise for the first time.

1.3. Injectivity, surjectivity, bijectivity

In this section⁹, we recall some basic properties of maps – specifically, what it means for a map to be injective, surjective and bijective. We begin by recalling basic definitions:

• The words “map”, “mapping”, “function”, “transformation” and “operator”

are synonyms in mathematics.¹⁰

• A map f : X →Ybetween two sets X andY is said to beinjectiveif it has the following property:

– Ifx1andx2are two elements ofXsatisfying f (x1) = f (x2), thenx1 =x2. (In words: If two elements of X are sent to one and the same element of Y by f, then these two elements of X must have been equal in the first place. In other words: An element of X is uniquely determined by its image under f.)

Injective maps are often called “one-to-one maps” or “injections”.

For example:

– The map Z → _Z, x 7→ 2x (this is the map that sends each integer x to 2x) is injective, because ifx1andx2are two integers satisfying 2x1 =2x2, then x₁= x₂.

– The mapZ→_Z, x 7→ x²(this is the map that sends each integerxtox²) is notinjective, because if x₁ and x₂ are two integers satisfying x²₁ = x²₂, then we do not necessarily have x₁ = x2. (For example, if x₁ = −1 and x2 =1, thenx₁²= x²₂ but notx1 =x2.)

a map, because the map that it is supposed to define (i.e., the map fromNtoZ which sends every x∈Nto x

1+x) does not exist (forx=2, we have x

1+x = ²

1+2 ∈/Z, which shows that a map fromNtoZcannot send thisxto this x

1+x). Thus, when defining a map fromXtoY (using whatever notation), do not forget to check that it is well-defined (i.e., that your definition specifies precisely one image for each x∈ X, and that these images all lie inY). In many cases, this is obvious or very easy to check (I will usually not even mention this check), but in some cases, this is a difficult task.

9a significant part of which is copied from [Grinbe16b, §3.21]

10That said, mathematicians often show some nuance by using one of them and not the other.

However, we do not need to concern ourselves with this here.

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• A map f : X → Y between two sets X and Y is said to be surjective if it has the following property:

– For each y∈ Y, there exists some x ∈ _X _satisfying _f(x) = y. (In words:

Each element ofY is an image of some element ofX under f.) Surjective maps are often called “onto maps” or “surjections”.

For example:

– The map Z → _Z, x 7→ x+1 (this is the map that sends each integer x to x+1) is surjective, because each integeryhas some integer satisfying x+1=y (namely, x =y−1).

– The map Z → _Z, _x 7→ _2x (this is the map that sends each integer x to 2x) is not surjective, because not each integer y has some integer x satisfying 2x=y. (For instance,y=1 has no suchx, sincey is odd.) – The map {1, 2, 3, 4} → {1, 2, 3, 4, 5}, x 7→ x (this is the map sending

each xtox) isnotsurjective, because not eachy∈ {1, 2, 3, 4, 5}has some x∈ {1, 2, 3, 4} satisfying x=y. (Namely, y=5 has no suchx.)

• A map f : X → Y between two sets X and Y is said to be bijective if it is both injective and surjective. Bijective maps are often called “one-to-one correspondences” or “bijections”.

For example:

– The map Z → _Z, x 7→ x+1 is bijective, since it is both injective and surjective.

– The map{1, 2, 3, 4} → {1, 2, 3, 4, 5}, x7→ xisnotbijective, since it is not surjective. (However, it is injective.)

– The map Z → _N, x 7→ |x| is not bijective, since it is not injective.

(However, it is surjective.)

– The mapZ →_Z, x7→ x² isnotbijective, since it is not injective. (It also is not surjective.)

• If X is a set, then idX denotes the map from X to X that sends each x ∈ X tox itself. (In words: idX denotes the map which sends each element of X to itself.) The map id_X is often called the identity map on X, and often denoted by id (when X is clear from the context or irrelevant). The identity map idX

is always bijective.

• If f : X → Y and g : Y → Z are two maps, then the composition g◦ f of the maps g and f is defined to be the map from X to Z that sends each x ∈ X to g(f (x)). (In words: The composition g◦ f is the map from X to Z that applies the map f first and then applies the map g.) You might find it confusing that this map is denoted by g◦ f (rather than f ◦g), given that it proceeds by applying f first and g last; however, this has its reasons:

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It satisfies (g◦ f) (x) = g(f (x)). Had we denoted it by f ◦g instead, this equality would instead become (f ◦g) (x) = g(f (x)), which would be even more confusing.

• If f : X → Y is a map between two sets X and Y, then an inverseof f means a map g : Y → X satisfying f ◦ g = idY and g◦ f = idX. (In words, the condition “f ◦g = _id_Y” means “if you start with some element y ∈ Y, then apply g, then apply f, then you get y back”, or equivalently “the map f undoes the mapg”. Similarly, the condition “g◦ f =idX” means “if you start with some element x ∈ X, then apply f, then apply g, then you get x back”, or equivalently “the mapg undoes the map f”. Thus, an inverse of f means a mapg :Y →X that both undoes and is undone by f.)

The map f : X →Yis said to beinvertibleif and only if an inverse of f exists.

If an inverse of f exists, then it is unique¹¹, and thus is calledthe inverse of f, and is denoted by f⁻¹.

For example:

– The map Z→_Z, x 7→ x+1 is invertible, and its inverse isZ→_Z, x 7→

x−_1.

– The map Q\ {₁} → Q\ {₀}, x 7→ ¹

1−x is invertible, and its inverse is the mapQ\ {₀} →Q\ {₁}, x 7→₁−¹

x.

• If f : X →Yis a map between two sets XandY, then the following notations will be used:

– For any subset U of X, we let f(U) be the subset {f (u) | u∈U} of Y. This set f(U) is called the image of U under f. This should not be

11Proof.Letg1andg2be two inverses of f. We shall show thatg1=g2.

We know thatg1is an inverse of f. In other words,g1is a mapY→Xsatisfying f ◦g1=idY

andg1◦f =idX.

We know thatg₂is an inverse of f. In other words,g₂is a mapY→Xsatisfying f ◦g₂=_id_Y andg₂◦f =_id_X_.

A well-known fact (known asassociativity of map composition, and stated explicitly as Propo- sition 2.82 below) says that if X,Y, Zand W are four sets, and ifc : X → Y, b : Y → Z and a:Z→Ware three maps, then

(a◦b)◦c=a◦(b◦c).

Applying this fact to Y, X, Y, X, g1, f and g2 instead of X, Y, Z, W, c, b and a, we obtain (g₂◦f)◦g₁=g₂◦(f ◦g₁).

Hence, g₂◦(f◦g₁) = (g₂◦f)

| {z }

=id_X

◦g₁ = id_X◦g₁ = g₁. Comparing this with g₂◦(f ◦g₁)

| {z }

=id_Y

= g₂◦id_Y=g₂, we obtaing₁=g₂.

Now, forget that we fixedg₁and g₂. We thus have shown that if g₁ and g₂ are two inverses of f, then g₁ = g2. In other words, any two inverses of f must be equal. In other words, if an inverse of f exists, then it is unique.

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confused with the image f (x) of a single elementx ∈ X under f.

Note that the map f : X→Y is surjective if and only ifY= f(X). (This is easily seen to be a restatement of the definition of “surjective”.)

– For any subsetV ofY, we let f⁻¹(V)be the subset{u ∈ X | f (u)∈ V} of X. This set f⁻¹(V) is called the preimage of V under f. This should not be confused with the image f⁻¹(y) of a single element y∈ _Y _under the inverse f⁻¹ of f (when this inverse exists).

(Note that in general, f f⁻¹(V) 6= V and f⁻¹(f (U)) 6= U. However, f f⁻¹(V)⊆V and U ⊆ f⁻¹(f(U)).)

– For any subset U of X, we let f |_U be the map fromU toY which sends eachu ∈U to f (u) ∈Y. This map f |_U is called therestrictionof f to the subset U.

The following facts are fundamental:

Theorem 1.1. A map f : X →Yis invertible if and only if it is bijective.

Theorem 1.2. LetU andV be two finite sets. Then,|U| =|V|if and only if there exists a bijective map f : U →V.

Theorem 1.2 holds even if the setsU andV are infinite, but to make sense of this we would need to define the size of an infinite set, which is a much subtler issue than the size of a finite set. We will only need Theorem 1.2 for finite sets.

Let us state some more well-known and basic properties of maps between finite sets:

Lemma 1.3. LetU and V be two finite sets. Let f : U →V be a map.

(a)We have |f(S)| ≤ |S|for each subset SofU.

(b)If |f (U)| ≥ |U|, then the map f is injective.

(c)If f is injective, then |f (S)|=|S| for each subsetSofU.

Lemma 1.4. Let U and V be two finite sets such that |U| ≤ |V|. Let f : U → V be a map. Then, we have the following logical equivalence:

(f is surjective) ⇐⇒ (f is bijective).

Lemma 1.5. Let U and V be two finite sets such that |U| ≥ |V|. Let f : U → V be a map. Then, we have the following logical equivalence:

(f is injective) ⇐⇒ (f is bijective).

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Exercise 1.1. Prove Lemma 1.3, Lemma 1.4 and Lemma 1.5.

Let us make one additional observation about maps:

Remark 1.6. Composition of maps is associative: If X, Y, Z and W are three sets, and if c : X → Y, b : Y → Z and a : Z → W are three maps, then (a◦b)◦c =a◦(b◦c). (This shall be proven in Proposition 2.82 below.)

In Section 2.13, we shall prove a more general fact: If X₁,X2, . . . ,X_k+1 are k+1 sets for some k ∈ _{N, and if} fi : Xi → Xi+₁ is a map for each i ∈ {1, 2, . . . ,k}, then the composition f_k ◦ f_k₋₁◦ · · · ◦ f₁ of all k maps f₁, f₂, . . . ,f_k is a well-defined map from X₁ to X_k+1, which sends each element x ∈ X₁ to f_k(f_k−1(f_k−2(· · ·(f2(f1(x)))· · ·))) (in other words, which transforms each element x ∈ X₁ by first applying f₁, then applying f₂, then applying f₃, and so on); this composition f_k ◦ f_k−1 ◦ · · · ◦ f₁ can also be written as f_k ◦ (f_k−1◦(f_k−2◦(· · · ◦(f2◦ f1)· · ·))) or as (((· · ·(f_k◦ f_k−1)◦ · · ·)◦ f3)◦ f2)◦ f1. An important particular case is when k = 0; in this case, f_k◦ f_k₋₁◦ · · · ◦ f₁ is a composition of 0 maps. It is defined to be id_X₁ (the identity map of the setX₁), and it is called the “empty composition of maps X1 → X1”. (The logic behind this definition is that the composition f_k◦ f_k₋₁◦ · · · ◦ f₁ should transform each element x ∈ X₁ by first applying f₁, then applying f2, then applying f3, and so on; however, for k = 0, there are no maps to apply, and so x just remains unchanged.)

1.4. Sums and products: a synopsis

In this section, I will recall the definitions of the∑and∏signs and collect some of their basic properties (without proofs). When I say “recall”, I am implying that the reader has at least some prior acquaintance (and, ideally, experience) with these signs; for a first introduction, this section is probably too brief and too abstract.

Ideally, you should use this section to familiarize yourself with my (sometimes idiosyncratic) notations.

Throughout Section 1.4, we letAbe one of the sets N, Z,Q, Rand C.

1.4.1. Definition of ∑

Let us first define the∑sign. There are actually several (slightly different, but still closely related) notations involving the∑sign; let us define the most important of them:

• If S is a finite set, and if as is an element of A for each s ∈ S, then ∑

s∈S

as

denotes the sum of all of these elements as. Formally, this sum is defined by recursion on|S|, as follows:

– If |S|=0, then ∑

s∈S

as is defined to be 0.

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– Let n∈ N. Assume that we have defined ∑

s∈S

as for every finite setSwith

|S| = n (and every choice of elements as of A). Now, if S is a finite set with |S| = n+1 (and if as ∈ A are chosen for all s ∈ S), then ∑

s∈S

as is defined by picking any t ∈S ¹² and setting

s

∑

∈S

as =at+

∑

s∈S\{t}

as. (1)

It is not immediately clear why this definition is legitimate: The right hand side of (1) is defined using a choice of t, but we want our value of

s∑∈S

as to depend only onS and on the as (not on some arbitrarily chosen t ∈ S). However, it is possible to prove that the right hand side of (1) is actually independent of t (that is, any two choices of t will lead to the same result). See Section 2.14 below (and Theorem 2.118(a)in particular) for the proof of this fact.

Examples:

– If S ={4, 7, 9} and as = ¹

s² for every s ∈ _{S, then} _∑

s∈S

as = a4+a7+a9 = 1

4² + ¹ 7² + ¹

9² = ⁶⁰⁴⁹ 63504.

– If S = {1, 2, . . . ,n} (for some n ∈ _{N) and} as = s² for every s ∈ S, then

s∑∈S

as = _∑

s∈S

s² = 1²+2²+· · ·+n². (There is a formula saying that the right hand side of this equality is 1

6n(2n+1) (n+1).) – If S=_∅, then ∑

s∈S

as =0 (since |S|=0).

Remarks:

– The sum ∑

s∈S

as is usually pronounced “sum of the as over all s ∈ S” or

“sum of the as with s ranging over S” or “sum of the as with s running through all elements of S”. The letter “s” in the sum is called the “summation index”¹³, and its exact choice is immaterial (for example, you can rewrite ∑

s∈S

as as ∑

t∈S

at or as ∑

Φ∈S

a_Φ or as ∑

♠∈S

a♠), as long as it does not already have a different meaning outside of the sum¹⁴. (Ultimately,

12This is possible, becauseSis nonempty (in fact,|S|=n+1>n≥0).

13The plural of the word “index” here is “indices”, not “indexes”.

14If it already has a different meaning, then it must not be used as a summation index! For example, you must not write “everyn∈_Nsatisfies ∑

n∈{0,1,...,n}

n= ⁿ(n+1)

2 ”, because here the summation indexnclashes with a different meaning of the lettern.

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a summation index is the same kind of placeholder variable as the “s”

in the statement “for all s ∈ S, we have as+2as = 3as”, or as a loop variable in a for-loop in programming.) The sign ∑ itself is called “the summation sign” or “the ∑sign”. The numbers as are called theaddends (or summands) of the sum ∑

s∈S

as. More precisely, for any givent ∈ S, we can refer to the number at as the “addend corresponding to the index t”

(or as the “addend fors =t”, or as the “addend fort”) of the sum ∑

s∈S

as. – When the set S is empty, the sum ∑

s∈S

as is called an empty sum. Our definition implies that any empty sum is 0. This convention is used throughout mathematics, except in rare occasions where a slightly subtler version of it is used¹⁵. Ignore anyone who tells you that empty sums are undefined!

– The summation index does not always have to be a single letter. For instance, ifSis a set of pairs, then we can write ∑

(x,y)∈S

a₍_x,y₎ (meaning the same as ∑

s∈S

as). Here is an example of this notation:

∑

(x,y)∈{1,2,3}²

x y = ¹

1 +¹ 2+¹

3 +² 1+²

2 +² 3+³

1 +³ 2+³

3 (here, we are using the notation ∑

(x,y)∈S

a(x,y) with S = {1, 2, 3}² and a(x,y) = ^x

y). Note that we could not have rewritten this sum in the form

s∑∈S

as with a single-letter variableswithout introducing an extra notation such as a₍_x,y₎ for the quotients x

y.

– Mathematicians don’t seem to have reached an agreement on the operator precedence of the ∑ sign. By this I mean the following question:

15Do not worry about this subtler version for the time being. If you really want to know what it is: Our above definition is tailored to the cases when the as are numbers (i.e., elements of one of the sets N,Z,Q,Rand C). In more advanced settings, one tends to take sums of the form

s∈S∑

as where theas are not numbers but (for example) elements of a commutative ringK. (See Definition 6.2 for the definition of a commutative ring.) In such cases, one wants the sum ∑

s∈Sas

for an empty set S to be not the integer 0, but the zero of the commutative ringK (which is sometimes distinct from the integer 0). This has the slightly confusing consequence that the meaning of the sum ∑

s∈Sas for an empty setSdepends on what ringKtheas belong to, even if (for an empty setS) there are noasto begin with! But in practice, the choice ofKis always clear from context, so this is not ambiguous.

A similar caveat applies to the other versions of the∑ sign, as well as to the ∏sign defined further below; I shall not elaborate on it further.

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Does ∑

s∈S

as+b(wherebis some other element ofA) mean ∑

s∈S

(as+b)or

s∑∈S

as

+b ? In my experience, the second interpretation (i.e., reading it as

s∑∈S

as

+b) is more widespread, and this is the interpretation that I will follow. Nevertheless, be on the watch for possible misunderstand- ings, as someone might be using the first interpretation when you expect it the least!¹⁶

However, the situation is different for products and nested sums. For instance, the expression ∑

s∈S

basc is understood to mean ∑

s∈S

(basc), and a nested sum like ∑

s∈S ∑

t∈T

as,t (whereSand T are two sets, and where as,t is an element ofAfor each pair(s,t)∈ S×T) is to be read as ∑

s∈S

t∑∈T

as,t

. – Speaking of nested sums: they mean exactly what they seem to mean.

For instance, ∑

s∈S ∑

t∈T

as,tis what you get if you compute the sum ∑

t∈T

as,t for each s∈ S, and then sum up all of these sums together. In a nested sum

s∑∈S ∑

t∈T

as,t, the first summation sign (∑

s∈S

) is called the “outer summation”, and the second summation sign (∑

t∈T

) is called the “inner summation”.

– An expression of the form “∑

s∈S

as” (whereSis a finite set) is called afinite sum.

– We have required the set S to be finite when defining ∑

s∈S

as. Of course, this requirement was necessary for our definition, and there is no way to make sense of infinite sums such as ∑

s∈_Zs². However, some infinite sums can be made sense of. The simplest case is when the set Smight be infinite, but only finitely many among theas are nonzero. In this case, we can define ∑

s∈S

as simply by discarding the zero addends and summing the finitely many remaining addends. Other situations in which infinite sums make sense appear in analysis and in topological algebra (e.g., power series).

– The sum ∑

s∈S

as always belongs toA. ¹⁷ For instance, a sum of elements ofNbelongs to N; a sum of elements ofRbelongs to R, and so on.

• A slightly more complicated version of the summation sign is the following:

Let S be a finite set, and let A(s) be a logical statement defined for every

16This is similar to the notorious disagreement about whethera/bcmeans(a/b)·cora/(bc).

17Recall that we have assumedAto be one of the setsN,Z,Q,RandC, and that we have assumed theasto belong toA.

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s ∈ S ¹⁸. For example, S can be {1, 2, 3, 4}, and A(s) can be the statement

“s is even”. For each s ∈ S satisfying A(s), let as be an element of A. Then, the sum ∑

s∈S;

A(s)

as is defined by

s

∑

∈S;

A(s)

as =

∑

s∈{t∈S | A(t)}

as.

In other words, ∑

s∈S;

A(s)

as is the sum of the as for all s ∈S which satisfy A(s). Examples:

– If S = {1, 2, 3, 4, 5}, then ∑

s∈S;

sis even

as = a₂+a₄. (Of course, ∑

s∈S;

sis even

as is

s∈∑S;

A(s)

as when A(s)is defined to be the statement “sis even”.)

– If S = {1, 2, . . . ,n} (for some n ∈ _{N) and} as = s² for every s ∈ S, then

s∈∑S;

sis even

as =a2+a4+· · ·+a_k, where kis the largest even number among 1, 2, . . . ,n(that is,k =nif nis even, andk =n−1 otherwise).

Remarks:

– The sum ∑

s∈S;

A(s)

as is usually pronounced “sum of theas over alls ∈ Ssatis- fyingA(s)”. The semicolon after “s ∈ S” is often omitted or replaced by a colon or a comma. Many authors often omit the “s ∈ S” part (so they simply write ∑

A(s)

as) when it is clear enough what the Sis. (For instance, they would write ∑

1≤s≤5

s² instead of ∑

s∈_N;

1≤s≤5

s².) – The set S needs not be finite in order for ∑

s∈S;

A(s)

as to be defined; it suffices that the set {t∈ S | A(t)} be finite (i.e., that only finitely many s ∈ S satisfy A(s)).

– The sum ∑

s∈S;

A(s)

as is said to be empty whenever the set {t ∈S | A(t)} is empty (i.e., whenever no s∈ Ssatisfies A(s)).

18Formally speaking, this means thatAis a map fromSto the set of all logical statements. Such a map is called apredicate.

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• Finally, here is the simplest version of the summation sign: Letuandvbe two integers. We agree to understand the set {u,u+1, . . . ,v} to be empty when u>v. Let as be an element ofAfor each s∈ {u,u+1, . . . ,v}. Then, ∑^v

s=uas is defined by

∑

v s=u

as =

∑

s∈{u,u+1,...,v}

as. Examples:

– We have ∑⁸

s=3

1

s = _∑

s∈{3,4,...,8}

1 s = ¹

3+¹ 4 +¹

5+¹ 6 +¹

7+¹

8 = ³⁴¹ 280. – We have ∑³

s=3

1

s = _∑

s∈{3}

1 s = ¹

3. – We have ∑²

s=3

1 s = _∑

s∈_∅

1 s =0.

Remarks:

– The sum ∑^v

s=uas is usually pronounced “sum of the as for all s from u to v (inclusive)”. It is often written au+au+₁+· · ·+av, but this latter notation has its drawbacks: In order to understand an expression like au+a_u+1+· · ·+av, one needs to correctly guess the pattern (which can be unintuitive when the as themselves are complicated: for example, it takes a while to find the “moving parts” in the expression 2·7

3+2 + 3·7

3+3 +· · ·+ ⁷·7

3+7, whereas the notation ∑⁷

s=₂

s·7

3+s for the same sum is perfectly clear).

– In the sum ∑^v

s=uas, the integer u is called the lower limit (of the sum), whereas the integer v is called the upper limit (of the sum). The sum is said tostart(orbegin) atuand endatv.

– The sum ∑^v

s=uas is said to be empty whenever u > v. In other words, a sum of the form ∑^v

s=uas is empty whenever it “ends before it has begun”.

However, a sum which “ends right after it begins” (i.e., a sum ∑^v

s=uaswith u=v) is not empty; it just has one addend only. (This is unlike integrals, which are 0 whenever their lower and upper limit are equal.)

– Let me stress once again that a sum ∑^v

s=uas with u > v is empty and equals 0. It does not matter how much greater u is than v. So, for

Notes on the combinatorial fundamentals of algebra

fundamentals of algebra ∗

Darij Grinberg May 25, 2021

(with minor corrections May 25, 2021)

Contents

1. Introduction

1.1. Prerequisites

1.2. Notations

1.3. Injectivity, surjectivity, bijectivity

1.4. Sums and products: a synopsis

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fundamentals of algebra ^∗