1. Syntax clarification.

Syntax clarification.

I don’t mean to be splitting hairs but these things I am wondering about.

Is this formally acceptable as a definition of “f”, f:N->R, f(x)=2x

Or, must one write: f:N->R, f|->2x

Would one ever write: f:N->R, f(x)|->2x

True or False

1. It is improper to say "the function f(x)”.
2. It is proper to say that the function “f” is defined by f(x) =2x.
3. It is improper to say that the function “f” is defined by f(x).
4. It is proper to say that the function “f” is defined by f|-> 2x.
5. It is proper to refer to “f(x)” as the image of “f”.
6. It is proper to refer to “2x” as the image of “f”.
7. It is proper to refer to f(x) = 2x as the “characteristic function" of “f”.
7a. It is proper to refer to f(x) = 2x as the “characteristic equation" of “f”.
8. It is proper to refer to f|-> 2x as the “characteristic function of “f”.
8a. It is proper to refer to f|-> 2x as the “characteristic equation of “f”.
9. It is proper to say that N is the domain set of “f”.
10. It is improper to say that R is the range set of “f”.
11. It is proper to say that R is the codomain set of “f”.

My guess is that all of the foregoing are true, but I wouldn't bet even a small pig let alone the farm.

2. Re: Syntax clarification.

Originally Posted by Ray1234
Syntax clarification.

I don’t mean to be splitting hairs but these things I am wondering about.

Is this formally acceptable as a definition of “f”, f:N->R, f(x)=2x
Yes, that is perfectly valid. It is a little "unusual" in that, while the range of f certainly is a subset of R, writing f:N->N (or even f:N->E, the set of even integers) would be more accurate.

Or, must one write: f:N->R, f|->2x
I would consider this incorrect since "x" is not specified on the left. Perhaps f: x|-> 2x would work.

Would one ever write: f:N->R, f(x)|->2x
I wouldn't. I would interpret "|->" as meaning "goes to" or "changes to" and f(x) is already 2x.
Again, I would prefer "x|-> 2x".

True or False

1. It is improper to say "the function f(x)”.
Strictly speaking that is false but it is a commonly used "abuse of notation".

2. It is proper to say that the function “f” is defined by f(x) =2x.
With the unstated understanding that the domain of f is the largest set for which that formula makes sense, yes. (Here that would normally be either the real numbers or complex numbers depending upon the "domain of discourse".) Strictly speaking a function is defined by both a formula and a domain.

3. It is improper to say that the function “f” is defined by f(x).
No, that wouldn't make any sense to me.

4. It is proper to say that the function “f” is defined by f|-> 2x.
With the understanding that "f|->2x" (as above I would prefer f:x|-> 2x) implies that f(x)= 2x and the same understanding about the domain as above, yes, that is acceptable.

5. It is proper to refer to “f(x)” as the image of “f”.
No, it is not. The "image" of a function is the set of all values of f. I would accept { f(x)| x in whatever domain f has} as the image. Personally, I have always preferred f(A)= {f(x)| x $\in$ A} as the "image of f on A" with the set A specified.

6. It is proper to refer to “2x” as the image of “f”.
Same answer as above with, I presume, f(x)= 2x.

7. It is proper to refer to f(x) = 2x as the “characteristic function" of “f”.
I have never seen "characteristic function" used in that way! Why talk about the characteristic function of a function?

7a. It is proper to refer to f(x) = 2x as the “characteristic equation" of “f”.
No!

8. It is proper to refer to f|-> 2x as the “characteristic function of “f”.
No, that's simply an improper use of the phrase "characteristic function" (in English. Are you translating this from another language? Perhaps "characteristic" is not the right word.)

8a. It is proper to refer to f|-> 2x as the “characteristic equation of “f”.
No. Same as above with the added objection that "f|-> 2x" is not an "equation"!

9. It is proper to say that N is the domain set of “f”.
First, I would not say "domain set", I would just say "domain". The "domain" of a function is already a set. Second, whether it is proper or not would depend upon whether or not the set of all natural numbers was the domain of f! Are we still talking about f:N->R, f(x)= 2x?

10. It is improper to say that R is the range set of “f”.
Different texts may say different things about this. Some will say that the range of f:N-> R, f(x)= 2x has range R, others would say the range is the set of all positive, even integers. Personally, I would say that R is the range and E, the set of positive even integers, is the "image".

11. It is proper to say that R is the codomain set of “f”.
No. The codomain is the same as the image.

My guess is that all of the foregoing are true, but I wouldn't bet even a small pig let alone the farm.

3. Re: Syntax clarification.

A couple of followup questions if I may, but first:

regarding, f:N->R, f|->2x instead of f:N->R, x|->2x, that one was just a flub reproduced by copy and paste, alas that really muddied everything up.

Regarding f:N->R, f(x)|->2x, that was me being misdirected by my own flub. (alas)2

1) I had not thought about it but I take your point about the accuracy of reducing the codomain from R to N or even E for the function defined by f(x) = 2x.

I had not run across E = the set of even integers. Does O = the set of odd integers? Hmmm, an internet indicates that E and O are informal and should defined before being used ... which is what you did.

2) Regarding: It is proper to refer to “f(x)” as the image of “f”.

Answer: No. OK then, given f:N->N, x|->2x, is it proper to regard f(x), (or, 2x), as a specific element of the image of f (since f(x) is NOT, properly speaking, the "function f(x)" nor it's image)?

Hmmm, I really wanted f(x) to be shorthand for the image of f, is there anyway of expressing the image of f given that the context, say f:N->N, x|->2x, is already declared, something like [f(x)] or {f(x)}, incidentally do those expressions have standard meanings?

3) Regarding "image", "range", "codomain". I would like to be sure that I have this correct.

The codomain is the the set from which correspondences to domain elements are drawn. The "range" is the subset of the codomain elements that actually have a correspondence to at least one domain element. The "image" is the same thing as the "range". In the statement g:N->L, L (whatever it is) is the codomain (but some texts might interpret L as the range).

I am not sure if I got that right.

4) Regarding. Characteristic function\equation.

OK, equation is out, no question.

After reviewing the meaning of a "characteristic function" I see that this is not what I was thinking ... out.

Given that f:N->N, x|->2x is first declared, one can say:

a) f(x) = 2x is the defining function of f.

b) f is defined by f(x)=2x.

c) f(x) = 2x is the defining property of f.

d) f(x) = 2x is the characteristic property of f.

vis-a vis:

Let (a_1, b_1) and (a_2, b_2) be ordered pairs. Then the characteristic (or defining) property of the ordered pair is:

(a_1, b_1) = (a_2, b_2) if and only if a_1 = a_2 and b_1 = b_2.

I cannot really find a definition of a "characteristic property" other than what is implied by the above example. Perhaps the use of the words "property" and "characteristic property" are reserved for definitions involving classes of objects?

Thank you.

4. Re: Syntax clarification.

Originally Posted by Ray1234

A couple of followup questions if I may, but first:

regarding, f:N->R, f|->2x instead of f:N->R, x|->2x, that one was just a flub reproduced by copy and paste, alas that really muddied everything up.

Regarding f:N->R, f(x)|->2x, that was me being misdirected by my own flub. (alas)2

1) I had not thought about it but I take your point about the accuracy of reducing the codomain from R to N or even E for the function defined by f(x) = 2x.

I had not run across E = the set of even integers. Does O = the set of odd integers? Hmmm, an internet indicates that E and O are informal and should defined before being used ... which is what you did.
The difference between "co-domain" and "range" is often too subtle for beginners to fully grasp. It starts mattering more when we start talking about properties of (classes of) functions, rather than individual functions.

However, it is (somewhat) important to make this distinction, because the function $f: A \to f(A)$ is surjective, while $f:A \to B$ may well not be. So why do we consider such things?

Well, for example, if one has a real-valued function, it may not be obvious what set its range actually IS, but it may be clear that whatever it is, it IS some set of real numbers. It's often quite a lot of work to determine the range of a function.

2) Regarding: It is proper to refer to “f(x)” as the image of “f”.

Answer: No. OK then, given f:N->N, x|->2x, is it proper to regard f(x), (or, 2x), as a specific element of the image of f (since f(x) is NOT, properly speaking, the "function f(x)" nor it's image)?
There is a difference between "an" image of $f$ (a value $f(a)$, given a domain element $a$, which is the image of $a$ under $f$) and an image-set (a set of such image values). If you think of "image" as being used like so:

The image of _____ under $f$, then which of these two you have depends on what KIND of thing (subset of the domain, or single element of the domain) you put in the blank.

Hmmm, I really wanted f(x) to be shorthand for the image of f, is there anyway of expressing the image of f given that the context, say f:N->N, x|->2x, is already declared, something like [f(x)] or {f(x)}, incidentally do those expressions have standard meanings?
The image of $f$ usually means $f(\text{domain}(f))$, so if $f:A \to B$, then $\text{im }f = f(A)$. If you want a specific element of $f(A)$, you have to add: the image of $x$ is $f(x)$...see the difference?

3) Regarding "image", "range", "codomain". I would like to be sure that I have this correct.

The codomain is the the set from which correspondences to domain elements are drawn. The "range" is the subset of the codomain elements that actually have a correspondence to at least one domain element. The "image" is the same thing as the "range". In the statement g:N->L, L (whatever it is) is the codomain (but some texts might interpret L as the range).

I am not sure if I got that right.
That seems OK.

4) Regarding. Characteristic function\equation.

OK, equation is out, no question.

After reviewing the meaning of a "characteristic function" I see that this is not what I was thinking ... out.

Given that f:N->N, x|->2x is first declared, one can say:

a) f(x) = 2x is the defining function of f.
I would use the term "defining rule".

b) f is defined by f(x)=2x.
Close, but no cigar. You need to specify the domain and co-domain as well.

c) f(x) = 2x is the defining property of f.
No, "property" isn't quite the word you're looking for. Again, "rule" or even "equation" would be better. You want to indicate that $f = \{(x,2x): x \in \text{domain}(f)\}$, that is, that $x$ has image $2x$.

d) f(x) = 2x is the characteristic property of f.
Characteristic property has other meanings that would confuse your readers. See above.

vis-a vis:

Let (a_1, b_1) and (a_2, b_2) be ordered pairs. Then the characteristic (or defining) property of the ordered pair is:

(a_1, b_1) = (a_2, b_2) if and only if a_1 = a_2 and b_1 = b_2.

I cannot really find a definition of a "characteristic property" other than what is implied by the above example. Perhaps the use of the words "property" and "characteristic property" are reserved for definitions involving classes of objects?

Thank you.
A "characteristic property" means informally, a typical property. It's a bit vague. "Characterizing property" is a property that essentially defines something, for example:

"k is an integer, less than 4, and greater than 2" is a characterizing property of 3, in that we can deduce which integer we mean by the property stated. Again, I would argue for clearer and simpler language, to reduce confusion.

*******************

In my opinion, the most unambiguous way to define a function is:

$f:A \to B$ (read "f goes from A to B") where $f(a) = (\text{some expression})$, or $a \mapsto (\text{some expression})$ (read "a maps to _____").

So, for example, the squaring function on the natural numbers is:

$f: \Bbb N \to \Bbb N$, where $f(k) = k^2$

-OR-

$f: \Bbb N \to \Bbb N$, where $k \stackrel{f}{\mapsto} k^2$.

We could restrict the co-domain (or TARGET set) to the set of square positive integers, but an EXPLICIT listing of that set would require calculating every square (which would take....a long time). Some mathematicians will insist that such a restriction of the co-domain yields a distinct function, and some mathematicians "don't care". The behavior of $f$ on the domain is going to be the same, and we get the same set of ordered pairs in $f$ either way. In lay terms, taking a larger co-domain corresponds to having "a taller space for the graph, even though some/most of it might be empty".

Functions can be QUITE complicated, even though this is not apparent from the definition. For example, the function that associates with every pair (m,n) of natural numbers the m-th decimal digit of the n-th power of $\pi$ (in decimal form) is, to my knowledge, at any rate, fairly unpredictable, even though the range is clearly just a subset of {0,1,2,3,4,5,6,7,8,9}.

5. Re: Syntax clarification.

OK, excellent (again). I appreciate your clarity. I regularly circle back to review your replies, and the replies of others. I almost always see things in a new light and learn something new. Thanks.