Mathematical Definitions: If versus Iff

[quantumdude]

Mathematical Definitions: "If" versus "Iff"

One might expect, perhaps naively, that mathematical definitions should be interpreted as "iff" statements by default, unless stated otherwise. This expectation arises from the facts that a definition expresses an equivalence, and that equivalences are biconditional.

But the more I read mathematics textbooks, the more it seems that the distinction between "if" and "iff" is not respected. Consider for example the following definition from Elementary Linear Algebra, 5ed, by Larson, Edwards, and Falvo (Houghton Mifflin Co.).

Definition of a Diagonalizable Matrix

An $n \times n$ matrix $A$ is diagonalizable if $A$ is similar to a diagonal matrix. That is, $A$ is diagonalizable if there exists an invertible matrix $P$ suth that

$$P^{-1}AP$$

is a diagonal matrix.

My question is: Why are those ifs not iffs instead? Are the authors just being sloppy?

 

[Hurkyl]

It's supposed to be iff. I suppose it's just tradition or something. I've been mentally substituting "iff" for a long time, so I no longer notice when "if" is used.

 

[matt grime]

In a case like that the 'only if' part is given for free, since a diagonal matrix is similar to a diagonal matrix and hence diagonalizable.

 

[quantumdude]

It's supposed to be iff.

That's what I thought.

I've been mentally substituting "iff" for a long time, so I no longer notice when "if" is used.

That brings me to my next question: How do you know when you can make the substitution?

Consider this example from Calculus, 8ed, by Larson, Hostetler, and Edwards (Houghton Mifflin Co.).

Definition of Continuity at a Point

A function $f$ is continuous at c if the following three conditions are met.

1. $f(c)$ is defined.

2. $\lim_{x\rightarrow c}f(x)$ exists.

3. $\lim_{x\rightarrow c}f(x)=f(c)$.

In that definition, it is supposed to be "if". Otherwise you would conclude that the function $f(x)=\sqrt{1-x^2}$ is not continuous at $x=\pm 1$ because the limit doesn't exist at either of those points.

 

[matt grime]

In case you didn't see it owing to time reasons see my reply: that is an if and only if statement for necessarily reversible reasons.

 

[shmoe]

It's pretty normal to use "if" in mathematical definitions. It's more along the lines of how "if" is used in everyday speech than in logic, such as "a motorcycle is called a crotch rocket if it goes over 225 km/h and causes severe back pains". You don't then call a harley a crotch rocket.

Sadly that was the only example that's coming to me right now.

 

[shmoe]

In that definition, it is supposed to be "if". Otherwise you would conclude that the function $f(x)=\sqrt{1-x^2}$ is not continuous at $x=\pm 1$ because the limit doesn't exist at either of those points.

This is a sloppy definition if it's supposed to encompass the case of endpoints. I would hope they seperately mention the case of endpoints.

Of course they may have already mentioned something about the meaning of a "two sided" limits at points where the function is defined on only one "side", so this could still be included (I'm not sure I've ever seen an interpretation like this though).

 

[quantumdude]

This is a sloppy definition if it's supposed to encompass the case of endpoints. I would hope they seperately mention the case of endpoints.

They define continuity on a closed interval (with its attendant one-sided limits) a couple of pages later.

This was first brought to my attention a year ago when a student asked me about the continuity of a function like $f(x)=\sqrt{1-x^2}$ at $x=\pm 1$. He mentally substituted "iff" in for "if" in the definition I quoted, and concluded that $f(x)$ is not continuous at the endpoints. I pointed out that he shouldn't be using the two-sided limit. I subsequently wrote up his argument as a "find the flaw in this reasoning" problem and have been assigning it ever since.

 

[JasonRox]

Maybe you should avoid books by Larson and Edwards.

 

[quantumdude]

I would love to ditch these books. But they are picked by my Department Chair, and I am a lowly Adjunct Instructor.

 

[shmoe]

They define continuity on a closed interval (with its attendant one-sided limits) a couple of pages later.

No problem with the book then, that definition of continuity is for points contained in an open interval that's contained in the domain of f. Your student tried to apply that definition where it didn't apply.

 

[quantumdude]

The calc book I don't have much of an issue with. I do think it should make a point of the use of "if" in that definition, but since they don't, I do it in class. The student was definitely the one in error, but the mistake was a subtle one.

But I do think that the linear algebra book should have used iff in the definition of a diagonalizable matrix.

 

[Hurkyl]

It's an "iff".

There's a horrible implicit thing being applied here: if you're claiming $f(x) := \sqrt{1 - x^2}$ is a function, then you had better be working in the topological space [-1, 1]. If you're working in the topological space [-1, 1], $\lim_{x \rightarrow 1} f(x)$ does equal f(1). Since there does not exist points greater than 1, the ordinary "two-sided" limit at 1 is exactly the same thing as the left one-sided limit.

And, if you were using some generalized definition of function so that this f was a "function" on the reals, then I would agree with your student that said this function is not continuous at 1. (when working over the reals) Of course, I'm assuming what it would even mean for such a generalized function to have a limit.

 

[shmoe]

But I do think that the linear algebra book should have used iff in the definition of a diagonalizable matrix.

I would still maintain it's a different usage of the word "if" than you find in logic say. To elaborate on what I mean with a less silly example, we call a tree deciduous "if" it sheds it's leaves before winter. Would you call a pine tree deciduous? In the same way I wouldn't call a matrix that's not similar to any diagonal matrix diagonalizable if I had read your books definition.

It's also the language that can be found in definitions by many respected authors, Titchmarsh, Rudin, Lang, Royden, and speaking of a gold standard in linear algebra, Hoffman & Kunze. In fact I had to do a bit of searching on my shelf before I found a definition that used "iff", that was Ahlfor's complex analysis which oddly used "iff" for the definition of a compact space but none of the other definitions I looked at. Maybe because it was followed by a few "iff" equivalences of compact that he had iff on the brain.

Regarding the limit, I'd prefer if they looked at limits as being in whatever topological space makes sense for the domain as Hurkyl is talking about, but this isn't the kind of viewpoint you're likely to get from a "stock" calculus text.

 

[quantumdude]

It's an "iff". There's a horrible implicit thing being applied here: if you're claiming $f(x) := \sqrt{1 - x^2}$ is a function, then you had better be working in the topological space [-1, 1]. If you're working in the topological space [-1, 1], $\lim_{x \rightarrow 1} f(x)$ does equal f(1).

The problem was indeed framed that way. The exact problem statement was:

Discuss the continuity of the function $f(x)=\sqrt{25-x^2}$ on the interval $[-5,5]$,

which is similar enough to the $f(x)$ that I used.

Since there does not exist points greater than 1, the ordinary "two-sided" limit at 1 is exactly the same thing as the left one-sided limit.

That part is news to me. Would I find it in a book on analysis? Topology? Either?

Fortunately it hasn't been an issue, because I steer them away from applying this definition of continuity for functions on closed intervals anyway.

And, if you were using some generalized definition of function so that this f was a "function" on the reals, then I would agree with your student that said this function is not continuous at 1.

You may have agreed with the conclusion, but you certainly would not have accepted his argument. He said that the function is discontinuous at the endpoints because the limit doesn't exist there. Taking the definition exactly as it appears in the textbook, the student definitely committed the fallacy of denying the antecedent.

But even if we replace "if" with "iff", the argument still fails because the limit does in fact exist.

 

[quantumdude]

I would still maintain it's a different usage of the word "if" than you find in logic say.

That is news to me too.

To elaborate on what I mean with a less silly example, we call a tree deciduous "if" it sheds it's leaves before winter. Would you call a pine tree deciduous? In the same way I wouldn't call a matrix that's not similar to any diagonal matrix diagonalizable if I had read your books definition.

Yes, but in the "real world" example there is common sense at work. What I am wondering is when we should do that in mathematics.

Just to put this into perspective: My graduate education is in theoretical physics, but I am currently teaching 1st and 2nd year undergrad math (intro linear algebra and below). I was not sure of how much liberty I could take with my interpretation of definitions. More specifically, I was under the impression that I could take no liberties with them. So I held that "if" means the logical "if", and nothing else. But now I have a better picture of things, I think.

It's also the language that can be found in definitions by many respected authors, Titchmarsh, Rudin, Lang, Royden, and speaking of a gold standard in linear algebra, Hoffman & Kunze.

OK, so it's normal. That's one of the things I was trying to find out.

Regarding the limit, I'd prefer if they looked at limits as being in whatever topological space makes sense for the domain as Hurkyl is talking about, but this isn't the kind of viewpoint you're likely to get from a "stock" calculus text.

I would prefer it too. I will definitely be reading up on this and finding a way to teach it in a calculus course.

Thanks to everyone for helping to clear this up.

 

[shmoe]

There has to be some agreed upon English (or other language) or you are going to have trouble communicating the math. You could probably do it, but it would involve a lot of grunting and frantic waving

About the limit thing, you're just changing the "if 0<|x-a|<delta" part of the limt to "if 0<|x-a|<delta and x is in the domain of f". have a look at how limits are defined in metric spaces. You'd just be considering the closed interval a metric space, and you can't "see" points outside this interval when looking at limits.

In single variable calc, they do everything in terms of left and right limits. This is a pretty natural thing to do on the real line, but having a more general notion of the limit introduced in single variable would probably make the jump to higher dimensions seem a little smaller.

 

[Hurkyl]

But even if we replace "if" with "iff", the argument still fails because the limit does in fact exist.

I would argue that the limit does, in fact, not exist.

The problem here is that we've not defined our terms! We all learned in calculus the definition of the limit of a function at a point. But we have not (at least I have not) ever seen anyone define the limit of a partial function[/b] at a point.

I presume that our difference in opinion is that we have different opinions on the "proper" definition of the limit of a partial function.


The definition I had in mind of $\lim_{x \rightarrow a} f(x) = L$ (on the reals) is:

$$\forall \epsilon > 0 : \exists \delta > 0 : \forall x \in (a - \delta, a) \cup (a, a + \delta) : |f(x) - L| < \epsilon$$

Thus, the expression $\lim_{x \rightarrow 1} \sqrt{1 - x^2}$ is ill-defined because we can never choose $\delta$ so that $f(x)$ is known to exist.



I'm not saying this is the right way to do this; better ideas spring to mind, but I'm not entirely sure I like their consequences either.

 

[quantumdude]

I would argue that the limit does, in fact, not exist.

No, I'm talking about the problem that I quoted. We were restricted to the space between the endpoints. You said that in that case, the limit as x approaches an endpoint exists and is equal to the value of the function there.

 

[shmoe]

No, I'm talking about the problem that I quoted. We were restricted to the space between the endpoints. You said that in that case, the limit as x approaches an endpoint exists and is equal to the value of the function there.

It would really depend on how the book defined $$\lim_{x\rightarrow a} f(x)$$. It's typical for a calculus text to use the usual epsilon-delta way that makes no limitations of the "x" being in the domain of f(x) (what Hurkyl gave in post 18). However, when they do this they will (or should!) explicitly state that this definition of the limit is for values of a where f is defined on an open interval containing a.

This is a little bad in the sense that for some deltas you may be considering "invalid x" values, but if delta is small enough you're ok given that f is defined in a neighbourhood of a, so it's not a huge deal.

But your text gave a different definition for continuity at an endpoint so they likely will only talk about one sided limits at endpoints.

 

[Doodle Bob]

But the more I read mathematics textbooks, the more it seems that the distinction between "if" and "iff" is not respected. Consider for example the following definition from Elementary Linear Algebra, 5ed, by Larson, Edwards, and Falvo (Houghton Mifflin Co.).
My question is: Why are those ifs not iffs instead? Are the authors just being sloppy?

I would say that relatively few mathematical texts use "iff" when making definitions. There are two factors involved with this. First has to do with English language conventions. It is conventional simply to use "if" with the understanding that with mathematical definitions there is an equivalence involved. Use of the phrase "if and only if" would be handier and obviously more mathematically correct.

The second has to do with the texts you're looking at. They are both written to an entry-level audience with most likely a large group of non-math majors, who will not understand the phrase "iff" at all and in fact most likely attribute it to a typo and ignore it.

As for myself, I really dislike "iff" in printed word (it's lazy and elitist) but use it all the time in the classroom.

 

[matt grime]

I think you need to closely examine the definition of continuity in the book given for consistency. Obviously the book greatly abuses the notion of function: f(x)=sqrt(1-x^2) needs its domain and range specifying, once we restrict to [-1,1] it is a prefectly continuous function at the end points; it makes no sense to speak of one sided butnot two sided continuity since you cannot approach from outside the interval, ie you need to think in the subspace topology. You can't say something fails because of behaviour outside its area of definition.

I always prefer when and only when; it is easier to explain than if and only if, somehow it is clearer when trying to prove a when and only when statement what direction you're going in at any time.

 

[robert Ihnot]

Hurkyl: There's a horrible implicit thing being applied here: if you're claiming is a function, then you had better be working in the topological space [-1, 1]. If you're working in the topological space [-1, 1], does equal f(1). Since there does not exist points greater than 1, the ordinary "two-sided" limit at 1 is exactly the same thing as the left one-sided limit.

Actually the definition is gone over at: http://archives.math.utk.edu/visual.calculus/1/continuous.5/index.html
where the author uses iff, and only later explicitly clarifies this for endpoints.

Even so, author says: $$lim x\longrightarrowa+$$ is used for a left hand endpoint, whereas I was somehow under the impression that the + meant we went through larger and larger values, but here it is used in the opposite way.