(A new question of the week)
A set is closed under an operation if, whenever that operation is applied to two elements of the set, the result is still an element of the set. It’s straightforward … until you look closely at some details! In the course of the discussion, we’ll dig into different definitions for division, and subtleties in the definition of closed sets.
The problem
We got this question from an algebra tutor, Michael, in late February:
I have a question about the closure of sets under different operations, dating back to something I saw in an old Modern Algebra textbook. The problem asked to state whether the set {0} is closed under each of addition, subtraction, multiplication, and division.
So, clearly it’s closed under addition, subtraction, and multiplication, but what about division? The answer given in the textbook says that it is not closed because 0÷0 is undefined. That was my first inclination as well, but when I thought about it some more, I found a more fundamental confusion. When a nonzero dividend is divided by zero, the answer does not exist. But when the dividend is zero, the answer is indeterminate. So, if my replacement set is a singleton set consisting of 0 only, does that not resolve the issue of its indeterminacy?
I asked another math teacher, who responded that because 0÷0 is indeterminate, it could equal a number that is outside the set. I understand that issue, but it doesn’t resolve the question for me. Seems to me, if the domain under consideration contains one unique value that satisfies the operation, then whatever other values which exist outside of that domain are irrelevant.
Is there something more fundamental to the definition of closure that I’m misunderstanding here? Or am I correct about this?
Thanks,
Michael
A set A is closed under an operation * if, for any two elements a and b of A, a * b is an element of A. For example, the set of positive integers is closed under addition because the sum of any two positive integers is still a positive integer. I think of it as saying that you can’t “escape” from the set using the operation. Here, we are asking about an unusually small set, the set consisting of only the element zero. Since \(0+0\), \(0-0\), and \(0\times 0\) are all zero, it is closed under addition, subtraction, and multiplication. But what about division?
For information about indeterminate expressions, which we’ll be discussing further, see
Zero Divided By Zero: Undefined and Indeterminate
A formal answer
Doctor Fenton answered first:
Hi Michael,
If I use the letter S to denote one of the standard number systems, i.e. the natural numbers N (which includes 0 for this discussion), the integers Z, the rationals Q, the reals R, and the complex numbers C, all of the operations (addition, subtraction, and multiplication) are defined as functions on the Cartesian product S×S into S. However, division is defined, for Q, R, and C, not by a function on S×S, but only on S×(S – {0}), by (x, y)→x/y = x * y-1, where * denotes multiplication and y-1 is the multiplicative inverse of y. The multiplicative inverse is only defined on S – {0}.
The element (0, 0) belongs to each of N, Z, Q, R, and C, and the corresponding operation on {0}×{0} is just the restriction of that operation on S. But (0, 0) does not belong to the domain of the division function on Q, R, or C, so it is very different from the other operations. So while you could define a function from {0}×{0}→{0} and call it “division”, to me that would be misleading, since it suggests that it is also a restriction of an operation on the other number systems.
Actually, there is only one possible binary operation on {0}×{0}→{0}, and it has already been called addition, subtraction, and multiplication. You can give it as many names as you like, but what’s the point?
In case you are not familiar with all the notation used here, the Cartesian product \(A\times B\) means the set of all ordered pairs \((a,b)\) where a and b are elements of A and B, respectively. Addition, subtraction, and multiplication are all defined for any pair of numbers; so, for instance, when we define them on the integers, \(\mathbb{Z}\), they are defined as functions from \(\mathbb{Z}\times\mathbb{Z}\) to \(\mathbb{Z}\). But division does not allow b, the second argument, to be zero, so it is only defined on \(\mathbb{Z}\times(\mathbb{Z}-\{0\})\), which excludes pairs like \((4,0)\) or \((0,0)\).
And division is defined as multiplication by the reciprocal, or multiplicative inverse (except for integers, where the reciprocal is not defined, so a different definition is needed, which applies only to certain pairs – we’ll get to that later). The multiplicative inverse of a number a is the number b such that \(ab = 1\); that is, \(b = \frac{1}{a}\). There is no reciprocal for 0, so we can’t divide by it.
When we use an operation on a subset of any other set (e.g. addition on the positive integers, a subset of the integers), the resulting operation is a restriction of the operation as defined on the larger set – we don’t redefine addition of positive integers to be something different from what it is on all integers, but use the same operation while restricting it to elements of the subset. The operation of addition that we say is closed on the positive integers is the same operation we talk about for integers, just restricted to our smaller set.
Another perspective
I had been thinking about the question, but wanted to let others answer before adding my thoughts, which focused on his mention of indeterminate division, and which were aimed a somewhat lower level.
I intentionally held off on answering this question, expecting Doctor Fenton (or anyone) to have some good ideas by looking at the big picture, which I might miss. I was right. I’ll now offer my thoughts, which reflect some of the same ideas but at a more basic level.
First, the question,
State whether the set {0} is closed under each of addition, subtraction, multiplication, and division,
implies that we are thinking of this as a subset of the integers, or real numbers, or something, and therefore to inherit the standard operations from those larger sets. We are not thinking of creating from scratch an operation to call “division” on the new set.
“Inherit” is not quite a standard term for this (Doctor Fenton’s term, “restriction”, is more accurate), but I think it conveys my idea well: though we are looking at a smaller set, we keep the same meaning for the operation, rather than inventing a new meaning for the same name.
Second, the fact that 0/0 is indeterminate is really just a particular reason not to define it: not because there is no appropriate result, but because there are too many. Your thought that in the subset we no longer have “too many”, so we could say the answer is 0, is interesting, but misses that fact that the division operation we are inheriting is not defined. You are implicitly defining a new operation on the subset, which is not the same as division.
That is, division by zero, whether the dividend is zero or not, is undefined. As explained in the blog post referred to above, something like \(1\div 0\) is undefined because there is no number x such that \(1 = 0\times x\), and thus no candidate for the role of quotient, while \(0\div 0\) is undefined because for any number x, \(0 = 0\times x\), so there is no unique quotient – too many job applicants to be able to pick one. In either case, we don’t define the division.
Observe also that here I am using a different definition of division than Doctor Fenton used, since his doesn’t apply to integers, as he mentioned. In any case, since the “inherited” operation that we are “restricting” to this set is not defined for 0, there is nothing to inherit!
Michael’s idea is interesting, however: He was saying that if we define division within the set \(\{0\}\) as the inverse of multiplication, then there is only one number you can multiply by zero to get zero (namely the only number we have, zero!), division is no longer indeterminate, and we can choose to define it! Nevertheless, we have to return to the basic idea:
Third, this is the essence of the closure concept: A subset is closed under an operation defined on the entire set when its application on a subset always yields an element of the subset. Since division by 0 (in any case) does not yield zero, the subset is not closed. Again, the key idea is that we have to be talking about the operation on the containing set, not redefining it.
So, good ideas, but the book had it right.
Again, these are mostly just different ways to say the same things Doctor Fenton said, but focusing on integers, which is where the idea of division as indeterminate is commonly introduced.
Responses
Michael responded, first to Doctor Fenton:
Thank you Doctor Fenton,
I think I see. The operation of division (and the others too, for that matter) requires the context of a number system in order to define it. Zero in a domain by itself still retains that context. As you put it, the operation of division is S×(S – {0}) for any of them.
I was getting stuck on side-stepping the idea that y-1 does not exist for this set by defining (x, y)→x/y = a, where * denotes multiplication and a is the unique solution to a * y = x. At first this reasoning seemed compelling to me, but the y-1 not existing remains a problem, relating to the more general number system issue as indicated above.
Well said. The second paragraph relates to the alternate definition I used for integers. That is what we will be focusing on moving forward.
Then he answered me:
Thanks again Doctor Peterson,
I was coming to this understanding just as you were typing up your own response. Much appreciated!
Textbook definitions may vary!
Then he continued with a third response:
One more thought on this. I’ve traced my earlier confusion back to different definitions of division that were printed in separate editions of the Dolciani texbook.
In the 1962/65 edition, the division operation is defined as the product of the dividend with the reciprocal of the divisor.
In the 1978 edition, the quotient is defined as the number whose product with the divisor is the dividend (albeit, only for nonzero divisors). “If b is not zero, the quotient a ÷ b is defined to be the number x whose product with b is a.”
I believe the earlier edition is more mathematically correct.
“The Dolciani textbook” refers to one of several textbooks written in the 1960’s (my own school days). You will notice that the first definition is Doctor Fenton’s, while the second is the one I used, which applies to all types of numbers, including integers.
I suspect, from the earlier mention of “an old Modern Algebra textbook”, that he is referring to the book “Modern Algebra: Structure and Method”, and not, as I might have expected, a book on abstract algebra, which is often described as “modern algebra” in textbook titles, at which level the notations used by Doctor Fenton are more typically found. I wish I had access to the book so I could check the context of the question more fully.
Actually, upon a more careful reading of the 1962 passage, it uses the same wording as the 1978 edition, buried in the paragraph, and the specifics of this definition are beside the point anyway. Both editions say, as a first condition, that division is not defined when the divisor is zero, and it’s quite clear that there is no number system for which it could be defined, as per your explanation.
Thanks again for shedding light on this matter for me.
I answered, explaining about the difference in the definitions, after checking one good source to see what definition they use:
You may have noticed that Doctor Fenton said,
division is defined for Q, R, and C, … by (x, y)→x/y = x * y-1, where * denotes multiplication and y-1 is the multiplicative inverse of y.
Since the integers, Z, do not have a multiplicative inverse, that definition is not applicable. The definition of division in MathWorld is the more general one:
Division is the inverse operation of multiplication, so that if
a×b = c,
then a can be recovered as
a = c÷b
as long as b ≠ 0. In general, division by zero is not defined since the ability to “invert” a×b = c to recover a breaks down if b=0 (in which case c is always 0, independent of a).
So as you say, the end result is the same either way.
Wikipedia, though it has a more complicated treatment that I couldn’t as easily quote, defines it similarly, starting with division of natural numbers. It’s worth observing that the fact that integers lack a multiplicative inverse is related to the fact that the integers (or natural numbers) are not closed under division.
Is there a conflict?
Michael wrote back, offering a variation of the MathWorld definition:
I went over this exercise with my student last night, and it was great to have a well-thought-out answer to the question she’d raised last time.
I’m still puzzling a bit over how the definition of division works, referring specifically to Z (and similarly N0) as mentioned above. Perhaps I’m wrong, but it seems to me that this boils down to a subtlety in that definition.
Definition 1: (My prior understanding.)
Division is the inverse operation of multiplication, so that if
a×b = c,
then a can be recovered as
a = c÷b
as long as a exists and is unique. In general, division by zero is not defined because…
Definition 2: (The true definition?)
Division is the inverse operation of multiplication, so that if
a×b = c,
then a can be recovered as
a = c÷b
as long as b ≠ 0. In general, division by zero is not defined because…
So this boils down to 2 questions.
1) Is the b = 0 case excluded no matter what by definition, or by virtue of the fact that the definition requires the quotient to exist and be unique? (And yes, I’m aware I altered the explicit wording appearing on MathWorld.)
2) Does this distinction even matter, insofar as my original question is concerned, or is the overriding consideration in the definition of number systems and/or closure for sets that would override my belief that, by Definition 1 for division, my original assumption would have been correct. (For instance, as a subset of the number system N0, other solutions may exist, even though they are not in our subset, so therefore we cannot say it’s closed.)
Not to be beating a dead horse, but I’m really trying to get to the very bottom of this question.
Thank you,
Michael
The second version of the definition is what I had copied from MathWorld. The first changes how the condition for the existence of the quotient is stated.
I replied,
First, I agree with your Definition 1; “exists and is unique” is a central part of the definition. The Definition 2 is just what I found to quote, not necessarily what I prefer in this context.
There is no difference in the implications, which means they are equivalent definitions. The difference is only that the former is clearer on the reason for the restriction to b ≠ 0, which you have explained rather than just stating.
I’m assuming you are using N0 to mean the non-negative integers, sometimes called Z*.
Notation for natural numbers and related sets varies considerably; I find N0 included in Wikipedia here, and other symbols in MathWorld here.
Answering the first question, about why b = 0 is excluded:
Starting with your Definition 1, b = 0 is excluded when a ≠ 0 because c doesn’t exist, and when a=0 because it is not unique. For that reason, when we just want to state the definition without defending it, we can just say b ≠ 0.
That is, the reason given explicitly in the first can be taken as the reason for making the restriction without explanation in the second.
Back to the main point
As to the second question, about whether this is relevant to the original question about {0},
I think the overriding factor is simply that in talking about closure, we are talking about the operation as defined in the superset. As Doctor Fenton said, “the corresponding operation on {0}x{0} is just the restriction of that operation on S”, or as I said, we “inherit the standard operations from those larger sets”. (His term is better than mine.)
To put it another way, by either of your two definitions, division by 0 is never defined on the superset, whatever that may be, so it still isn’t defined when we focus on the subset. As an example, in group theory, a subgroup is a set that is closed under the group operation, not under some other operation you invent for the subgroup.
We also talk about closure of an operation on a group (or other algebraic structure) that is not inherently a subset; in those cases, it just means that the operation is defined (as an element in the set) for every pair. But that is not what is involved in our question, because we come to {0} already knowing what division means.
He concluded:
Got it. So the “inherited” definition, both in terms of the requirements it specifies and the numbers to be considered under those requirements, remain in the realm of the superset number system. Makes sense, the more that I think about it. Otherwise, the numbers themselves as well as the operations would lose their meaning.
A good conclusion.