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The associative law of multiplication for three positive integers $a,b$ and $c$ can be proved$^1$ from the Commutative Law and the property of "Number of things" easily.
We can prove$^2$ the associative law of multiplication for four positive integers $a,b,c$ and $d$ from the associative law of of multiplication of three positive integers and the property of "Number of things". We can also prove$^2$ that the multiplication of three positive integers is independent of the arrangement of the individual positive integers in the expression.

  • Can we prove$^3$ associative law of multiplication for $n$ numbers formally in some way?
  • Can we prove that the multiplication $a_1a_2a_3\cdots a_n$ is independent of the order in which the individual numbers are arranged e.g. $a_1a_2a_3\cdots a_n=a_1a_na_2a_3\cdots a_{n-1}$.$(a_n \in \mathbb{N}$ for every n), that is the commutative law?

This portion is additional, added for the sake of clearness, may be ommited if seems to be a discussion
$^1$ $abc = c$ sums such as $(a + a + \cdots$ to $b$ terms).
By commutative law $(a + a + \cdots$ to $b$ terms)=$(b + b + \cdots$ to $a$ terms).
So $(ab)c=(ba)c$.
$(ba)$ is the sum of $b$ numbers, each of which is $a$.
$(ba)c$ is the sum of $c$ numbers, each of which is $(ba)$, that is $(ba)c=(ba+ba+ba+\cdots$ to $c$ terms). $(ba)$ in $(ba)c$ can be considered as a coloumn of $a$ numbers each of which is $b$, i.e. $(ba)c= \begin{bmatrix}b\\b\\b\\.\\.\\.\end{bmatrix}_{a\times 1}+ \begin{bmatrix}b\\b\\b\\.\\.\\.\end{bmatrix}_{a\times 1}+\begin{bmatrix}b\\b\\b\\.\\.\\.\end{bmatrix}_{a\times 1} \cdots \begin{bmatrix}b\\b\\b\\.\\.\\.\end{bmatrix}_{a\times 1}$ upto $c$ times.
This can be rearranged as:
$(ba)c= {\begin{bmatrix}{\begin{bmatrix}b&b&b&.&.&.\end{bmatrix}_{1\times c}}\\{\begin{bmatrix}b&b&b&.&.&.\end{bmatrix}_{1\times c}}\\{\begin{bmatrix}b&b&b&.&.&.\end{bmatrix}_{1\times c}}\\.\\.\\.{\begin{bmatrix}b&b&b&.&.&.\end{bmatrix}_{c\times 1}}\end{bmatrix}_{a\times 1}}={\begin{bmatrix}bc\\bc\\bc\\.\\.\\.\\bc\end{bmatrix}}_{a\times1}=(bc+bc+bc\cdots a$ times) =$(bc)a=a(bc).$
Although associative law for three numbers can also be proved by considering $(ab)c$ as a surface $(ab)$(having $b$ rows of 1's and $a$ coloumns of 1's) repeated $c$ times in the 3rd dimension but I cannnot put it in latex so I chose to prove it as I did.

$^{2}$ $abcd = (ab)(c)(d) = (ba)(c)(d)=bacd$
$abcd= (ab)(c)(d)=(c)(ab)(d)=cabd$
$abcd=(c)(ab)(d)=(c)(ba)(d)=cbad$ similarly all other orders can be formed.

$^3$ Please do not use set-theory to prove the associative law. I am using numbers for concrete things.

user103816
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    Note that the associative law for three numbers cannot be proved from the commutative law. There are associative systems which are not commutative - group theory provides many examples, or matrices. It is possible to prove forms of generalised associative law for $n$ elements from the basic associative law for three elements, amounting to the assertion that it does not matter how you arrange the brackets, so you may as well leave them out - provided the order of the elements stays the same.

    Note: the commutative law affects the basic order of the elements.

     – Mark Bennet Apr 12 '14 at 15:13
  • What of your definition of multiplication? $ab=\sum_{I=1}^b a$? This feels like proof by obfuscating notation more than anything else. – Stella Biderman Apr 12 '14 at 15:14
  • Mark, can you give a specific example? Matrices are associative but not commutative, and groups by definition are associative. – Stella Biderman Apr 12 '14 at 15:16
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    Please define "number". – Bill Dubuque Apr 12 '14 at 15:21
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    A simple operator which is commutative but not associative is the "average" binary operator $a\star b = \frac{a+b}{2}$. @StellaBiderman – Thomas Andrews Apr 12 '14 at 15:28
  • The key trick is to even define the $n$-term associative law, but once you have the tools to define it, you can prove it fairly directly by induction. – Thomas Andrews Apr 12 '14 at 15:30
  • See also the question Does commutativity imply associativity? – Bill Dubuque Apr 12 '14 at 15:31
  • You are speaking of "The associative law for three numbers $a,b$ and $c$", without further detail, but your example regards multiplication. Thus, I suppose you are asserting that it is possible to prove associative law for multiplication starting from commutative and associative laws for addition... – Mauro ALLEGRANZA Apr 12 '14 at 15:32
  • @MauroALLEGRANZA Basically I am proving every Law(for addition as well as multiplication) from the property of Number of things. In proving associative law(And any other law) of multiplication for three numbers we can directly use the property of Number of things, that is without using commutative law( I cannot put that in Latex). Rather quoting associative law of addition I am using the property of Number of things. – user103816 Apr 12 '14 at 16:49
  • @StellaBiderman Defintion of multiplication of positive integers in Mathematics same as that my definition is. The sum of b numbers each of which is a. Perhaps I am using some old notation(because I am reading an old book), sorry for obscurity. – user103816 Apr 12 '14 at 16:54
  • @BillDubuque I meant positive integers. I have edited my question. – user103816 Apr 12 '14 at 16:57
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    What does this "numbers of things" law state? – Stella Biderman Apr 12 '14 at 21:17
  • @StellaBiderman: The number of things in any group of distinct things is independent of the characters of these things, of the order in which they may be arranged in the group, and of the manner in which they may be associated with one another in smaller groups. $-$ "Number of things Law" – user103816 Apr 13 '14 at 03:59

2 Answers2

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Yes. Prove, by induction on $n$, that any product of $n$ factors, no matter how it's parenthesized, agrees with the product of the same factors, in the same order, parenthesized "to the left", i.e., $((\dots((a_1a_2)a_3)\dots a_{n-2})a_{n-1})a_n$. You already know the first nontrivial case, $n=3$, so you only need to do the induction step.

Andreas Blass
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  • I am seeking for proof of more general rule of multiplication e.g. $abcde=aebcd$. This cannot be proved by induction. – user103816 Apr 12 '14 at 16:44
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    @Anupam A rigorous formal proof is linked in my answer here, from George Bergman's note on Lang's Algebra. Yours is the special case of the commutative monoid of positive integers. – Bill Dubuque Apr 12 '14 at 17:42
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    Anupam, that equation demonstrates commutativity, not associativity. Can you explain why you think induction doesn't work? – Stella Biderman Apr 12 '14 at 21:19
  • Andreas, the link you have given is using set theory. Is there any way to prove without using set theory. – user103816 Apr 13 '14 at 04:05
  • @Anupam What link are you referring to? I put no link into my answer. – Andreas Blass Apr 13 '14 at 11:28
  • This :http://math.stackexchange.com/questions/691003/a-question-about-associativity-in-monoids/691478#691478 – user103816 Apr 13 '14 at 11:32
  • Anupam, that wasn't the answerer's link and is not necessary for his answer.What is you're problem with this proof? If an arbitrary parentheses is equal to the specified one, then by transitivity of equality any two are equal. – Stella Biderman Apr 13 '14 at 14:42
  • @JoshuaBiderman I do not know how to solve by induction. I know how induction is used to prove farmula's but I am not proficient in it. I cannot figure out $n \to n+1$ step. – user103816 Apr 15 '14 at 03:57
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    Is the down-vote because I answered the question as it stood at the time and didn't answer the part about commutativity that was added afterward? Or is there something wrong with my answer? – Andreas Blass Jan 24 '15 at 14:42
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To me, one problem here is that you have not rigorously defined multiplication. As far as I am concerned, anything that uses "..." is hand waving.

Also, what is your definition of "number of things"? Please give examples.

marty cohen
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  • The Number of things in a group is that property of the group which remains unchanged during every change in the group which does not destroy the separateness of the things from one another or their common separateness from all other things. Such changes may be changes in the characteristics of the things or in their arrangement within the group. Again, changes of arrangement may be changes either in the order of the things or in the manner in which they are associated with one another in smaller groups. – user103816 Sep 09 '14 at 15:21
  • In the above mentioned defintion Group refers to a collection of distinct objects(abstract/concrete). – user103816 Sep 09 '14 at 15:25
  • One more thing by multiplication I meant: $2\times 4=2+2+2+2$ not $4+4$, i.e $a \times b= $ adding $a$, $b$ times not adding $b$, $a$ times. For me it is merely a observational fact that $a \times b = b \times a$ – user103816 Sep 09 '14 at 15:40
  • Marty, see http://philosophy.stackexchange.com/questions/10622/what-do-we-mean-by-the-term-number-of-things (which was formerly posted on MSE and was migrated out). – Asaf Karagila Sep 12 '14 at 07:15