2. A closer look at induction
2.14. General commutativity for addition of numbers
2.14.1. The setup and the problem
Throughout Section 2.14, we let A be one of the sets N, Z, Q, R and C. The elements ofAwill be simply callednumbers.
There is an analogue of Proposition 2.82 for numbers:
Proposition 2.103. Let a, b and c be three numbers (i.e., elements of A). Then, (a+b) +c =a+ (b+c).
Proposition 2.103 is known as the associativity of addition (in A), and is funda-mental; its proof can be found in any textbook on the construction of the number system76.
In Section 2.13, we have used Proposition 2.82 to show that we can “drop the parentheses” in a composition fn◦ fn−1◦ · · · ◦ f1of maps (i.e., all possible complete parenthesizations of this composition are actually the same map). Likewise, we can use Proposition 2.103 to show that we can “drop the parentheses” in a sum a1+a2+· · ·+an of numbers (i.e., all possible complete parenthesizations of this sum are actually the same number). For example, ifa,b,c,dare four numbers, then the complete parenthesizations of a+b+c+dare
((a+b) +c) +d, (a+ (b+c)) +d, (a+b) + (c+d), a+ ((b+c) +d), a+ (b+ (c+d)),
and all of these five complete parenthesizations are the same number.
However, numbers behave better than maps. In particular, along with Proposi-tion 2.103, they satisfy another law that maps (generally) don’t satisfy:
Proposition 2.104. Let a and b be two numbers (i.e., elements of A). Then, a+b =b+a.
Proposition 2.104 is known as thecommutativity of addition(inA), and again is a fundamental result whose proofs are found in standard textbooks77.
Furthermore, numbers can always be added, whereas maps can only be com-posed if the domain of one is the codomain of the other. Thus, when we want to take the sum ofn numbers a1,a2, . . . ,an, we can not only choose where to put the parentheses, but also in what order the numbers should appear in the sum. It turns
76For example, Proposition 2.103 is proven in [Swanso20, Theorem 3.2.3 (3)] for the case when A = N; in [Swanso20, Theorem 3.5.4 (3)] for the case when A = Z; in [Swanso20, Theorem 3.6.4 (3)] for the case whenA=Q; in [Swanso20, Theorem 3.7.11] for the case whenA=R; in [Swanso20, Theorem 3.9.2] for the case whenA=C.
77For example, Proposition 2.104 is proven in [Swanso20, Theorem 3.2.3 (4)] for the case when A = N; in [Swanso20, Theorem 3.5.4 (4)] for the case when A = Z; in [Swanso20, Theorem 3.6.4 (4)] for the case whenA=Q; in [Swanso20, Theorem 3.7.11] for the case whenA=R; in [Swanso20, Theorem 3.9.2] for the case whenA=C.
out that neither of these choices affects the result. For example, if a,b,c are three numbers, then all 12 possible sums
(a+b) +c, a+ (b+c), (a+c) +b, a+ (c+b), (b+a) +c, b+ (a+c), (b+c) +a, b+ (c+a), (c+a) +b, c+ (a+b), (c+b) +a, c+ (b+a)
are actually the same number. The reader can easily verify this for three numbers a,b,c (using Proposition 2.103 and Proposition 2.104), but of course the general case (with n numbers) is more difficult. The independence of the result on the parenthesization can be proven using the same arguments that we gave in Section 2.13 (except that the◦ symbol is now replaced by +), but the independence on the order cannot easily be shown (or even stated) in this way.
Thus, we shall proceed differently: We shall rigorously define the sum ofn num-bers without specifying an order in which they are added or using parentheses.
Unlike the composition ofnmaps, which was defined for anordered listofnmaps, we shall define the sum ofnnumbers for afamilyofnnumbers (see the next subsec-tion for the definisubsec-tion of a “family”). Families don’t come with an ordering chosen in advance, so we cannot single out any specific ordering for use in the definition.
Thus, the independence on the order will be baked right into the definition.
Different solutions to the problem of formalizing the concept of the sum of n numbers can be found in [Bourba74, Chapter 1, §1.5]78 and in [GalQua20, §3.3].
2.14.2. Families
Let us first define what we mean by a “family” of n numbers. More generally, we can define a family of elements of any set, or even a family of elements ofdifferent sets. To motivate the definition, we first recall a concept of ann-tuple:
Remark 2.105. Letn ∈N.
(a)Let A be a set. Then, to specify ann-tuple of elements of Ameans specifying an element ai of A for each i ∈ {1, 2, . . . ,n}. This n-tuple is then denoted by (a1,a2, . . . ,an) or by(ai)i∈{1,2,...,n}. For each i∈ {1, 2, . . . ,n}, we refer to ai as the i-th entry of thisn-tuple.
The set of alln-tuples of elements of Ais denoted by An or by A×n; it is called the n-th Cartesian powerof the set A.
(b) More generally, we can define n-tuples of elements from different sets:
For each i ∈ {1, 2, . . . ,n}, let Ai be a set. Then, to specify an n-tuple of elements of A1,A2, . . . ,An means specifying an element ai of Ai for each i ∈ {1, 2, . . . ,n}. This n-tuple is (again) denoted by (a1,a2, . . . ,an) or by (ai)i∈{1,2,...,n}. For each i ∈ {1, 2, . . . ,n}, we refer toai as thei-th entry of thisn-tuple.
78Bourbaki, in [Bourba74, Chapter 1, §1.5], define something more general than a sum ofnnumbers:
They define the “composition” of a finite family of elements of a commutative magma. The sum of nnumbers is a particular case of this concept when the magma is the setA(endowed with its addition).
The set of all n-tuples of elements of A1,A2, . . . ,An is denoted by A1×A2×
· · · ×An or by ∏n
i=1
Ai; it is called theCartesian product of thensets A1,A2, . . . ,An. These nsets A1,A2, . . . ,An are called thefactorsof this Cartesian product.
Example 2.106. (a) The 3-tuple(7, 8, 9) is a 3-tuple of elements of N, and also a 3-tuple of elements ofZ. It can also be written in the form (6+i)i∈{1,2,3}. Thus, (6+i)i∈{1,2,3} = (6+1, 6+2, 6+3) = (7, 8, 9) ∈ N3 and also (6+i)i∈{1,2,3} ∈ Z3.
(b) The 5-tuple ({1},{2},{3},∅,N) is a 5-tuple of elements of the powerset of N(since{1},{2},{3},∅,Nare subsets of N, thus elements of the powerset ofN).
(c)The 0-tuple () can be viewed as a 0-tuple of elements ofanyset A.
(d) If we let [n] be the set {1, 2, . . . ,n} for each n ∈ N, then (1, 2, 2, 3, 3) is a 5-tuple of elements of [1],[2],[3],[4],[5] (because 1∈ [1], 2∈ [2], 2∈ [3], 3∈ [4] and 3∈ [5]). In other words, (1, 2, 2, 3, 3) ∈ [1]×[2]×[3]×[4]×[5].
(e) A 2-tuple is the same as an ordered pair. A 3-tuple is the same as an ordered triple. A 1-tuple of elements of a set A is “almost” the same as a single element of A; more precisely, there is a bijection
A→ A1, a7→ (a) from A to the set of 1-tuples of elements of A.
The notation “(ai)i∈{1,2,...,n}” in Remark 2.105 should be pronounced as “the n-tuple whose i-th entry is ai for each i ∈ {1, 2, . . . ,n}”. The letter “i” is used as a variable in this notation (similar to the “i” in the expression “∑n
i=1
i” or in the expression “the map N → N, i 7→ i+1” or in the expression “for all i ∈ N, we havei+1>i”); it does not refer to any specific element of{1, 2, . . . ,n}. As usual, it does not matter which letter we are using for this variable (as long as it does not already have a different meaning); thus, for example, the 3-tuples (6+i)i∈{1,2,3} and (6+j)j∈{1,2,3} and(6+x)x∈{1,2,3} are all identical (and equal (7, 8, 9)).
We also note that the “∏” sign in Remark 2.105(b)has a different meaning than the “∏” sign in Section 1.4. The former stands for a Cartesian product of sets, whereas the latter stands for a product of numbers. In particular, a product ∏n
i=1
ai of numbers does not change when its factors are swapped, whereas a Cartesian product ∏n
i=1
Ai of sets does. (In particular, if A and Bare two sets, then A×B and B×A are different sets in general. The 2-tuple (1,−1) belongs to N×Z, but not toZ×N.)
Thus, the purpose of ann-tuple is storing several elements (possibly of different sets) in one “container”. This is a highly useful notion, but sometimes one wants
a more general concept, which can store several elements but not necessarily or-ganized in a “linear order”. For example, assume you want to store four integers a,b,c,d in the form of a rectangular table
a b c d
(also known as a “2×2-table of integers”). Such a table doesn’t have a well-defined “1-st entry” or “2-nd en-try” (unless you agree on a specific order in which you read it); instead, it makes sense to speak of a “(1, 2)-th entry” (i.e., the entry in row 1 and column 2, which is b) or of a “(2, 2)-th entry” (i.e., the entry in row 2 and column 2, which is d).
Thus, such tables work similarly to n-tuples, but they are “indexed” by pairs (i,j) of appropriate integers rather than by the numbers 1, 2, . . . ,n.
The concept of a “family” generalizes both n-tuples and rectangular tables: It allows the entries to be indexed by the elements of an arbitrary (possibly infinite) set I instead of the numbers 1, 2, . . . ,n. Here is its definition (watch the similarities to Remark 2.105):
Definition 2.107. Let I be a set.
(a)Let Abe a set. Then, to specify an I-family of elements of Ameans specifying an element ai of A for each i ∈ I. This I-family is then denoted by (ai)i∈I. For each i ∈ I, we refer to ai as the i-th entry of this I-family. (Unlike the case of n-tuples, there is no notation like (a1,a2, . . . ,an) for I-families, because there is no natural way in which their entries should be listed.)
An I-family of elements of A is also called an A-valued I-family.
The set of all I-families of elements of A is denoted by AI or by A×I. (Note that the notation AI is also used for the set of all maps from I to A. But this set is more or less the same as the set of all I-families of elements of A; see Remark 2.109 below for the details.)
(b) More generally, we can define I-families of elements from different sets:
For each i ∈ I, let Ai be a set. Then, to specify an I-family of elements of (Ai)i∈I means specifying an element ai of Ai for each i ∈ I. This I-family is (again) denoted by(ai)i∈I. For eachi∈ I, we refer to ai as thei-th entryof this I-family.
The set of all I-families of elements of (Ai)i∈I is denoted by ∏
i∈I
Ai.
The word “I-family” (without further qualifications) means an I-family of el-ements of (Ai)i∈I for some sets Ai.
The word “family” (without further qualifications) means an I-family for some set I.
Example 2.108. (a) The family (6+i)i∈{0,3,5} is a {0, 3, 5}-family of elements of N (that is, an N-valued {0, 3, 5}-family). It has three entries: Its 0-th entry is 6+0 = 6; its 3-rd entry is 6+3 = 9; its 5-th entry is 6+5 = 11. Of course, this family is also a {0, 3, 5}-family of elements of Z. If we squint hard enough, we can pretend that this family is simply the 3-tuple (6, 9, 11); but this is not advisable, and also does not extend to situations in which there is no natural order on the set I.
(b)Let X be the set{“cat”, “chicken”, “dog”}consisting of three words. Then,
we can define an X-family (ai)i∈X of elements ofNby setting a“cat”=4, a“chicken” =2, a“dog” =4.
This family has 3 entries, which are 4, 2 and 4; but there is no natural order on the set X, so we cannot identify it with a 3-tuple.
We can also rewrite this family as
(the number of legs of a typical specimen of animali)i∈X.
Of course, not every family will have a description like this; sometimes a family is just a choice of elements without any underlying pattern.
(c)If Iis the empty set∅, and ifAis any set, then there is exactly one I-family of elements of A; namely, the empty family. Indeed, specifying such a family means specifying no elements at all, and there is just one way to do that. We can denote the empty family by (), just like the empty 0-tuple.
(d)The family(|i|)i∈Z is aZ-family of elements ofN(because|i|is an element ofNfor eachi ∈ Z). It can also be regarded as a Z-family of elements of Z.
(e) If I is the set {1, 2, . . . ,n} for some n ∈ N, and if A is any set, then an I-family (ai)i∈{1,2,...,n} of elements of A is the same as an n-tuple of elements of A.
The same holds for families and n-tuples of elements from different sets. Thus, any n setsA1,A2, . . . ,An satisfy ∏
i∈{1,2,...,n}
Ai = ∏n
i=1
Ai.
The notation “(ai)i∈I” in Definition 2.107 should be pronounced as “the I-family whose i-th entry is ai for each i ∈ I”. The letter “i” is used as a variable in this notation (similar to the “i” in the expression “∑n
i=1
i”); it does not refer to any specific element of I. As usual, it does not matter which letter we are using for this variable (as long as it does not already have a different meaning); thus, for example, the Z-families(|i|)i∈Z and(|p|)p∈Z and (|w|)w∈Z are all identical.
Remark 2.109. Let I and A be two sets. What is the difference between an A-valued I-family and a map from I to A? Both of these objects consist of a choice of an element of A for eachi ∈ I.
The main difference is terminological: e.g., when we speak of a family, the elements of A that constitute it are called its “entries”, whereas for a map they are called its “images” or “values”. Also, the notations for them are different:
The A-valued I-family (ai)i∈I corresponds to the map I → A, i7→ ai.
There is also another, subtler difference: A map from I to A “knows” what the set A is (so that, for example, the maps N → N, i 7→ i and N → Z, i 7→ i are considered different, even though they map every element of Nto the same value); but an A-valued I-family does not “know” what the set A is (so that, for example, the N-valued N-family (i)i∈N is considered identical with the Z-valued N-family (i)i∈N). This matters occasionally when one wants to consider
maps or families for different sets simultaneously; it is not relevant if we just work with A-valued I-families (or maps from I to A) for two fixed sets I and A. And either way, these conventions are not universal across the mathematical literature; for some authors, maps from ItoAdo not “know” whatAis, whereas other authors want families to “know” this too.
What is certainly true, independently of any conventions, is the following fact:
If I and A are two sets, then the map
{maps from I to A} → {A-valued I-families}, f 7→ (f (i))i∈I
is bijective. (Its inverse map sends every A-valued I-family (ai)i∈I to the map I → A, i 7→ ai.) Thus, there is little harm in equating {maps from I to A} with {A-valued I-families}.
We already know from Example 2.108 (e) that n-tuples are a particular case of families; the same holds for rectangular tables:
Definition 2.110. Let Abe a set. Let n∈ Nand m ∈ N. Then, ann×m-table of elements of Ameans an A-valued{1, 2, . . . ,n} × {1, 2, . . . ,m}-family. According to Remark 2.109, this is tantamount to saying that an n×m-table of elements of A means a map from {1, 2, . . . ,n} × {1, 2, . . . ,m} to A, except for notational differences (such as referring to the elements that constitute the n×m-table as
“entries” rather than “values”) and for the fact that an n×m-table does not
“know” A (whereas a map would do).
In future chapters, we shall consider “n × m-matrices”, which are de-fined as maps from {1, 2, . . . ,n} × {1, 2, . . . ,m} to A rather than as A-valued {1, 2, . . . ,n} × {1, 2, . . . ,m}-families. We shall keep using the same notations for them as for n×m-tables, but unlike n×m-tables, they will “know” A (that is, twon×m-matrices with the same entries but different sets Awill be considered different). Anyway, this difference is minor.
2.14.3. A desirable definition
We now know what an A-valued S-family is (for some set S): It is just a way of choosing some element of A for each s ∈ S. When this element is called as, the S-family is called (as)s∈S.
We now want to define the sum of an A-valued S-family (as)s∈S when the set S is finite. Actually, we have already seen a definition of this sum (which is called
s∑∈S
as) in Section 1.4. The only problem with that definition is that we don’t know yet that it is legitimate. Let us nevertheless recall it (rewriting it using the notion of anA-valued S-family):
Definition 2.111. If S is a finite set, and if (as)s∈S is an A-valued S-family, then we want to define the number ∑
s∈S
as. We define this number by recursion on|S| as follows:
• If|S|=0, then ∑
s∈S
as is defined to be 0.
• Let n ∈ N. Assume that we have defined ∑
s∈S
as for every finite set S with
|S| = n and any A-valued S-family (as)s∈S. Now, if S is a finite set with
|S| = n+1, and if (as)s∈S is any A-valued S-family, then ∑
s∈S
as is defined by picking anyt ∈ Sand setting
s
∑
∈Sas =at+
∑
s∈S\{t}
as. (161)
As we already observed in Section 1.4, it is not obvious that this definition is legitimate: The right hand side of (161) is defined using a choice oft, but we want our value of ∑
s∈S
as to depend only onSand(as)s∈S (not on some arbitrarily chosen t ∈ S). Thus, we cannot use this definition yet. Our main goal in this section is to prove that it is indeed legitimate.
2.14.4. The set of all possible sums
There are two ways to approach this goal. One is to prove the legitimacy of Defini-tion 2.111 by strong inducDefini-tion on|S|; the statementA(n)that we would be proving for eachn ∈Nhere would be saying that Definition 2.111 is legitimate for all finite setsS satisfying |S| = n. This is not hard, but conceptually confusing, as it would require us to use Definition 2.111 forsomesetsSwhile its legitimacy for other sets Sis yet unproven.
We prefer to proceed in a different way: We shall first define a set Sums (as)s∈S for any A-valued S-family (as)s∈S; this set shall consist (roughly speaking) of “all possible values that ∑
s∈S
ascould have according to Definition 2.111”. This set will be defined recursively, more or less following Definition 2.111, but instead of relying on a choice of some t ∈ S, it will use all possible elements t ∈ S. (See Definition 2.112 for the precise definition.) Unlike ∑
s∈S
as itself, it will be a set of numbers, not a single number; however, it has the advantage that the legitimacy of its definition will be immediately obvious. Then, we will prove (in Theorem 2.114) that this set Sums (as)s∈S is actually a 1-element set; this will allow us to define ∑
s∈S
as
to be the unique element of Sums (as)s∈S for anyA-valued S-family (as)s∈S (see Definition 2.116). Then, we will retroactively legitimize Definition 2.111 by showing
that Definition 2.111 leads to the same value of ∑
s∈S
as as Definition 2.116 (no matter whicht ∈ Sis chosen). Having thus justified Definition 2.111, we will forget about the set Sums (as)s∈S and about Definition 2.116.
In later subsections, we shall prove some basic properties of sums.
Let us define the set Sums (as)s∈S, as promised:
Definition 2.112. If S is a finite set, and if (as)s∈S is an A-valued S-family, then we want to define the set Sums (as)s∈S of numbers. We define this set by recursion on |S|as follows:
• If|S|=0, then Sums (as)s∈S is defined to be{0}.
• Let n ∈ N. Assume that we have defined Sums (as)s∈S for every finite set S with |S| = n and any A-valued S-family (as)s∈S. Now, if S is a finite set with |S| = n+1, and if (as)s∈S is any A-valued S-family, then Sums (as)s∈S is defined by
Sums (as)s∈S
=nat+b | t ∈ Sand b ∈Sums
(as)s∈S\{t}o. (162) (The sets Sums
(as)s∈S\{t} on the right hand side of this equation are well-defined, because for each t ∈ S, we have |S\ {t}| = |S| −1 = n (since|S| =n+1), and therefore Sums
(as)s∈S\{t}is well-defined by our assumption.)
Example 2.113. Let Sbe a finite set. Let (as)s∈S be an A-valuedS-family. Let us see what Definition 2.112 says whenS has only few elements:
(a)If S=∅, then
Sums (as)s∈∅={0} (163)
(directly by Definition 2.112, since |S|=|∅|=0 in this case).
(b)If S={x} for some element x, then Definition 2.112 yields Sums
(as)s∈{x}
=nat+b | t ∈ {x} and b∈ Sums
(as)s∈{x}\{t}o
=nax+b | b∈ Sums
(as)s∈{x}\{x}o (since the onlyt ∈ {x} is x)
=
ax+b | b∈ Sums (as)s∈∅
| {z }
={0}
(since {x} \ {x} =∅)
={ax+b | b ∈ {0}} ={ax+0}={ax}. (164)
(c)If S={x,y} for two distinct elementsx and y, then Definition 2.112 yields
(by (164), applied toyinstead ofx)
(d) Similar reasoning shows that if S = {x,y,z} for three distinct elements x, yand z, then
Sums
(as)s∈{x,y,z}=ax+ ay+az
,ay+ (ax+az),az+ ax+ay . It is not hard to check (using Proposition 2.103 and Proposition 2.104) that the three elements ax+ ay+az
Again, it is not hard to prove that ax+ ay+az+aw
=ay+ (ax+az+aw)
=az+ ax+ay+aw
= aw+ ax+ay+az , and thus the set Sums
(as)s∈{x,y,z,w} is again a 1-element set, whose unique element can be called ax+ay+az+aw.
These examples suggest that the set Sums (as)s∈Sshould always be a 1-element set. This is precisely what we are going to claim now:
Theorem 2.114. If S is a finite set, and if (as)s∈S is an A-valued S-family, then the set Sums (as)s∈S is a 1-element set.
2.14.5. The set of all possible sums is a 1-element set: proof
Before we step to the proof of Theorem 2.114, we observe an almost trivial lemma:
Lemma 2.115. Let a, b and c be three numbers (i.e., elements of A). Then, a+ (b+c) =b+ (a+c).
Proof of Lemma 2.115. Proposition 2.103 (applied tobandainstead ofaandb) yields (b+a) +c = b+ (a+c). Also, Proposition 2.103 yields (a+b) +c = a+ (b+c). Hence,
a+ (b+c) = (a+b)
| {z }
=b+a (by Proposition 2.104)
+c= (b+a) +c =b+ (a+c).
This proves Lemma 2.115.
Proof of Theorem 2.114. We shall prove Theorem 2.114 by strong induction on |S|: Let m∈ N. Assume that Theorem 2.114 holds under the condition that|S| <m.
We must now prove that Theorem 2.114 holds under the condition that |S|=m.
We have assumed that Theorem 2.114 holds under the condition that|S|<m. In other words, the following claim holds:
Claim 1: Let S be a finite set satisfying |S| < m. Let (as)s∈S be an A-valuedS-family. Then, the set Sums (as)s∈Sis a 1-element set.
Now, we must now prove that Theorem 2.114 holds under the condition that
|S| =m. In other words, we must prove the following claim:
Claim 2: Let S be a finite set satisfying |S| = m. Let (as)s∈S be an A-valuedS-family. Then, the set Sums (as)s∈Sis a 1-element set.
Before we start proving Claim 2, let us prove two auxiliary claims:
Claim 3: Let S be a finite set satisfying |S| < m. Let (as)s∈S be an A-valuedS-family. Letr ∈ S. Letg∈ Sums (as)s∈Sandc ∈ Sums(as)s∈S\{r}. Then, g=ar+c.
[Proof of Claim 3: The set S\ {r} is a subset of the finite set S, and thus itself is finite. Moreover,r ∈S, so that |S\ {r}|=|S| −1. Thus, |S| =|S\ {r}|+1. Hence, the definition of Sums (as)s∈S yields
Sums (as)s∈S =nat+b | t∈ S andb ∈ Sums(as)s∈S\{t}o. (165) But recall that r ∈ S and c ∈ Sums
(as)s∈S\{r}. Hence, the number ar+c has the form at+b for some t ∈ S and b ∈ Sums
(as)s∈S\{t} (namely, fort = r and b =c). In other words,
ar+c ∈ nat+b | t∈ S and b∈ Sums
(as)s∈S\{t}o. In view of (165), this rewrites asar+c ∈ Sums (as)s∈S.
But Claim 1 shows that the set Sums (as)s∈Sis a 1-element set. Hence, any two elements of Sums (as)s∈S are equal. In other words, any x ∈ Sums (as)s∈S and y ∈Sums (as)s∈S satisfy x =y. Applying this to x =g and y= ar+c, we obtain g = ar+c (since g ∈ Sums (as)s∈S and ar+c ∈ Sums (as)s∈S). This proves Claim 3.]
Claim 4: Let S be a finite set satisfying |S| = m. Let (as)s∈S be an A-valued S-family. Let p ∈ S and q ∈ S. Let f ∈ Sums
Claim 4: Let S be a finite set satisfying |S| = m. Let (as)s∈S be an A-valued S-family. Let p ∈ S and q ∈ S. Let f ∈ Sums