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Decide whether or not the following is a statement. In the case of a statement, say whether it is true or false: "Every real number is an even integer."

The above is question 1 from section 2.1 of Richard Hammack's introductory book Book of Proofs, and the author's answer is that the given sentence is false.

However, I think that, since "$x$ is an even integer" has a truth value only for integer $x$′s (for example, "$\sqrt{2}$ is an even integer" and "$\pi$ is an even integer" are each neither true nor false), the sentence "$\forall x {\in} \mathbb{R}\;x$ is an even integer" is not a statement, which in turn means that "Every real number is an even integer" is not a statement, let alone a false statement. Am I correct?

UPDATE

It's important that all use the same definition of even-ness. I will use the following: "An integer $x$ is even if $x=2k$ for some integer $k$".

My original Question had misquoted the exercise as

Decide whether or not the following is a statement. In the case of a statement, say whether it is true or false: "Every real number is even."

and I had kept saying "...is even" whenever I meant "...is an even integer."

There seems to be two opposing viewpoints in response to my initial post:

  1. Ethan Bolker argues that because even-ness is not defined for the real numbers, which it isn't assuming the above definition, then the sentence "Every real number is a even integer" isn't a statement as the definition can't be applied to it.

  2. fleablood argues that it is because of the fact that even-ness is not defined for the real numbers that the sentence "Every real number is a even integer" is false. I quote fleablood: "Indeed if A is only defined for objects which $X$ totally fails to be, that just confirms that $X$ is not $A$...".

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    $\begingroup$ I think it should be pointed out that Hammoks' "Book of Proofs" did not ask if "Every real number is even" is a statement. It asked if "Every real number is an even integer". $\endgroup$ Commented Jul 28 at 23:35
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    $\begingroup$ Very much so! Indicating whether something is or is not an even integer is always unambiguous. $6$ is an even integer. $\pi$ is not an even integer. Freddy the Penguin is not an even integer. The current tennis game is not an even integer. Saying something is "even" depends on what we mean. Is $\pi$ even? Is the current tennis game even? That depends on what we mean by "even" in that context. You point that "even" doesn't apply to non-integers so $\pi$ is even might not be a statement has some merit (I disagree but I could be wrong). But whether it is even integer is unambiguous. $\endgroup$ Commented Jul 29 at 1:23
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    $\begingroup$ @fleablood I would argue (somewhat pedantically) that the question "is the real number 1 an integer?" is ambiguous. (In the sense that there are reasonable interpretations of the question where the answer is yes, and others where the answer is no.) $\endgroup$ Commented Jul 29 at 4:37
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    $\begingroup$ @AeroMain27 In most foundational systems the technical answer to that question is false. Real numbers and integers are different types of things (or disjoint sets) and no integer is a real number or vice versa. What we have is an embedding $\mathbb Z\to \mathbb R$ and we can (informally, perhaps) call real numbers in the image of that embedding integers. But in usual colloquial math this is of course generally what we mean when we ask if a given real number is an integer. (And to your other question to fleablood, if I say the function $\cos(x)$ is even, do you feel the same way?) $\endgroup$ Commented Jul 29 at 5:06
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    $\begingroup$ -1, because as others have pointed out, the question misquotes the source. $\endgroup$ Commented Jul 29 at 15:50

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Ultimately, I think you are confusing a conditional in a definition to have the definition be true for an object, with a conditional for relevance or meaning.

A definition for "even integer" has an implied conditional that the object must be an integer for definition to be truthfully applied. But it is not a conditional for meaning.

"$\pi$ is not an integer" is a meaningful (and true) statement. So is "$\pi$ is not an integer of any sort" as is "$\pi$ is not any sort of integer and it is not any integer that happens to be even" and "$\pi$ is not an even integer" are all meaningful statements.

Hammack has gone to considerable effort to make his "domain of definition" to apply to all "things" so "My refrigerator is not an even integer" becomes a meaningful but false statement.

You on the other had a going to the other extremely that if a conditional is not achieved the definition isn't simply false, but meaningless. By your reckoning: "I am not an eagle" is meaningless. "A circle is not an isosceles triangle" is meaningless, because the condition "an eagle is a bird; I am not a bird" and "An isosceles triangle is a triangle; a circle is not a triangle". But these failures do not make the statements meaningless, it simply makes them false.

...

You had a valid observation when we were talking about "even" rather than "even integer" in that if the definition didn't apply to the the type of object it could be argued that the statements would be meaningless. It could be argued "$\pi$ is not prime" and "$\sqrt 7$ is not isosceles" and "$5+3 =$ the road to San Francisco" are meaningless. (But it could also be argued that they are just false.)

But Hammack goes out of his way to make his definitions apply to all "things".

"Freddy, the vampire penguin is not an even integer" is a true and valid statement as an "even integer" is a thing that is equal to $2$ times an integer. As Freddy the vampire penguin is not equal to $2$ times any integer it certainly can not be an even integer.

=======

I think it should be pointed out that Hammoks' "Book of Proofs" did not ask if "Every real number is even" is a statement. It asked if "Every real number is an even integer".

There may be some ambiguity about whether statements like "$\pi$ is even" or "The $\cos$ function is green" or "$2+3i > 3-2i$ (complex numbers)" are statements or not. (I'd personally say they are statements and false ones... although I'd demand a definition for "even"... and I'd be inclined to say "as order is not defined on the complex field '$2+3i > 3-2i$' can not be true and is therefore a false statement" but I wouldn't argue if someone countered with "As the context is meaningless it isn't a statement").

And Hammock's book is pretty basic and avoids the issue with "A statement is a sentence or a mathematical expression that is either definitely true or definitely false".

I'd say that is very clear we can ask if an every real number is an even integer and we can clearly answer "no".

So... a false statement.

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What is the truth value of "$\sqrt 2$ is even"

The truth value is "false"

or "$\pi$ is even".

The truth value is false.

I'd say that those sentences are neither true nor false and only for integer x′s can a sentence such as "x is even" have a truth value.

I'd say you are wrong.

$x$ is even means $x = 2k$ for some integer $k$. So if $x=2k$ for some integer $k$ then the truth value of $x=2k$ is TRUE and if $x$ is not equal to $2k$ for any integer $k$ the truth value is false.

So $\pi$ is not equal to $2k$ for any integer $k$ and $\sqrt 2$ is not equal to $2k$ for any integer $k$ and The Battle of New Orleans in 1815 is not equal to to $2k$ for any integer $2k$ then "$\pi$ is even", "$\sqrt 2$ is even", and "The Battle of New Orleans in 1815 is even" all have truth values of false.

For all the other x′s (i.e. x∈R−Z), "x is even" doesn't have a truth value.

As all $x \in \mathbb R\setminus \mathbb Z$ are not integers and all even numbers are integers then for each of those "$x$ is even" has truth value of FALSE.

Thus "∀x∈R, x is even" is not a statement

Yes, it is. It is a false statement.

, let alone a true one,

No. It's a false one.

and thus "every real number is even" is not a statement, let alone a true one.

It is a statement. It is a false one.

Am I correct in saying that "Every real number is even" is not a statement,

Nope.

let alone a true one?

It's certainly not a true statement.

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    $\begingroup$ I think the main issue with me accepting your is whether or not there is a difference between the following defintions even-ness. Definition 1: "An integer $x$ is even if $x=2k$ for some integer $k$. Definition 2: "$x$ is even if $x=2k$ for some integer $k$". The first definition seems to suggest that even-ness can only apply to integers whereas the second definition seems to suggest that even-ness can apply to any object. $\endgroup$ Commented Jul 29 at 5:03
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    $\begingroup$ But when an object is not in the domain of a definition, I don't think the definition can be even applied to the object and talking about a sentences like $X$ is a [new term introduced by the definition] or "$X$ is not a [new term introduced by the definition]" doesn't even make sense and can't be assigned truth values. $\endgroup$ Commented Jul 29 at 5:21
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    $\begingroup$ @AeroMain27: Hammack literally explains his definition as, "E equals the set of all things of form 2n, such that n is an element of $\Bbb Z$". Being an integer is actually not a prerequisite there (although it can be trivially deduced, of course). $\endgroup$ Commented Jul 29 at 14:34
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    $\begingroup$ The thing is the "domain" of the definition is not integers. It is any objects in the universe. A necessary (but not sufficient) requirement is that the object by an integer. But that's only a restriction as to when the definition is true. Not when the definition can be applied. There is nothing wrong with a definition of "even" as "even integer is a thing where a) it is an integer and 2) its divisible by $2$. Consider this: What is the truth value of "A circle is an isosceles triangle". Is the domain of "isosceles triagle" geometric figures, or triangles, or isosceles triangles? $\endgroup$ Commented Jul 29 at 14:59
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    $\begingroup$ ... That an "even integer" must be an integer is a "requirement to be true". It is not a "requirement to be defined". If you muddle the two we get sentences such as "$\pi$ is not an integer", and "A circle is not an isosceles triangle" and "I am not an eagle" as "meaningless" when they are all simply false statements. $\endgroup$ Commented Jul 29 at 15:10
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A correct answer to this question must deal with the ambiguity about whether and how "even" is defined for real numbers. If it's not then "Every real number is even" isn't a statement. If "even" is defined as "a multiple of 2" then "every real number is even" is a true statement. If "even" is defined as "twice an integer" then "every real number is even" is a false statement.

Daniel R. Collins comments that

Hammack's definition in Sec 1.1 is $E=\{2n:n \in \mathbb{Z}\}$

so Hammack is using the third alternative in this answer.

That alternative seems to be the one most other answerers and commenters here prefer. I lean toward "not a statement" as a better alternative psychologically, if not formally. If I were writing software packages to do arithmetic the predicate IsEven(x) would require that x be of type Integer. It would fail at compile time on IsEven(2.1)- the programming equivalent of "not a statement". (In object oriented languages IsEven()would be a method in class Integer.)

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    $\begingroup$ @TankutBeygu I don't think we disagree. The link you provide defines "even" only for integers. That's the first of the three possibly correct alternatives in my answer. $\endgroup$ Commented Jul 28 at 19:48
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    $\begingroup$ @TankutBeygu Please stop trolling this thread. $\endgroup$ Commented Jul 28 at 19:48
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    $\begingroup$ $1$ is not even. Who would choose a definition of even such that $1$ is even? $\endgroup$ Commented Jul 28 at 22:04
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    $\begingroup$ "Evenness and oddness are defined only for integers; for they make sense only for integers." So? To claim "X is not A" does not require that A apply to X. It just means that X fails to be A. Indeed, if A is only defined for object X totally fails to be that just confirms that X is not A. "$\pi$ is an integer" and "$\pi$ is an even integer" and "$\pi$ is even" are all valid statements by your definition of "even" and because "even" only applies to integers they are all false. $\endgroup$ Commented Jul 28 at 23:10
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    $\begingroup$ I meant "Indeed if A is only defined for objects which X totally fails to be, that just confirms that X is not A". Only animals can be carnivorous. My car is not an animal. So... my car is not carnivorous. Only integers can be even or odd. $\pi$ is not an integer. So $\pi$ is not even nor odd. "$\pi$ is not even" is a perfectly valid and true statement. $\endgroup$ Commented Jul 29 at 1:35
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It depends on the formal system you are using.

As other answers say, in set theory, this is usually considered a valid but false statement.

However, in some formal systems, particularly type theory, this is not a valid statement at all, so it's not possible to assert whether it is true or not.

In Martin-Lof type theory, for example, evenness can be defined as:

$\mathrm{even} = \lambda (n : \mathbb{N}). \sum_{m : \mathbb{N} } (m + m =_{\mathbb{N}} n)$

which has the type $\mathrm{even} : \mathbb{N} \rightarrow \mathbf{Type}$ (since propositions are represented as types in MLTT).

and the statement that "every real number is even" would be written as: $\prod_{x : \mathbb{R}} \mathrm{even}(x)$.

Since $\mathrm{even}$ has the type $\mathbb{N} \rightarrow \mathbf{Type}$, it can only take an argument that has the type $\mathbb{N}$, which $x$ does not. In MLTT, all terms and values are intrinsically typed. A statement that cannot be given a valid type is not a well-formed statement, and "every real number is even" is obviously not well-formed (it gives a function an argument of the wrong type), so one cannot talk about its truthfulness.

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    $\begingroup$ Although this answer is quite technical, I think it is the only one that actually addresses the underlying concerns in the question. $\endgroup$ Commented Jul 29 at 17:33
  • $\begingroup$ See also my answer here for an elaboration of why we could consider the statement to be false in set theory, rather than meaningless. $\endgroup$ Commented Jul 30 at 22:05
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I have taught from Hammack's book before, and I have disagreed with him on his answers about such questions, e.g., whether something is a statement or whether a statement is true or false. However, I still generally liked the book for my class, and I don't think this is really an issue.

The point is that the answer may depend upon the interpretation of the question, and there can be multiple reasonable interpretations. (In this case, the main two interpretations being: an even real number means an even integer and the property of being even is not defined for real numbers.) The important thing is to understand what are the reasonable interpretations, and what the answer is in under such interpretations. I tell my students their answers can also be right even if it's different from mine or the book's, but we should be able to understand how we interpreted things differently.

One possible misconception is the notion that there is a clear right or wrong answer in something like an introductory proof course. We try to approximate some ideal of objective correctness, but really there's always some interpretation, and the goal is to learn how to communicate in ways where we can (usually) interpret things in equivalent ways. There's not even a good clear definition of what a human proof is, and as others have commented, Hammack doesn't even try to give a precise definition of "statement."

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  • $\begingroup$ You saying that there is not a clear right or wrong answer does give me some comfort. The 2 popular perspectives so far (by fleablood and Ethan Bolker) are both perspectives that I can see being right. Maybe I should just be ok with saying that both answers are “right” in their own ways and just move on. I have spent a lot of time dealing with this question instead of actually freshening up my proof-writing skills which is the whole reason I started reading this book anyways. I’m starting to ask myself if spending more time on this question is even productive, maybe even counter-productive… $\endgroup$ Commented Jul 29 at 6:51
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    $\begingroup$ Note that it often takes more effort to be more precise, and when you do so you often make ideas less clear (especially to non-experts), so we have to balance precision and simplicity, sort of like an uncertainty principle. The looser you are on the precision side, the more logically consistent interpretations you allow. And often the writer doesn't think of all these interpretations, because they're so used to all the tacit assumptions they were making. $\endgroup$ Commented Jul 29 at 7:20
  • $\begingroup$ @AeroMain27 Yes, I think you should move on. But I've enjoyed the back and forth. $\endgroup$ Commented Jul 29 at 16:06
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    $\begingroup$ But the book defines an even integer (on page 5), and asserts that $\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R}$. The question "Decide whether or not the following are statements. In the case of a statement, say if it is true or false, if possible: 'Every real number is an even integer'" is unambiguous and false, in the context of that text. $\endgroup$ Commented Jul 30 at 1:08
  • $\begingroup$ @XanderHenderson Ah, my answer was about the original question: "Every real number is even". But I do remember there were exercises in Hammack there he asserted they were statements and I said they were not (something about Heaven or God comes to mind). $\endgroup$ Commented Jul 30 at 2:32
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A natural definition of even number is the following:

A real number $x$ is even if there exists an integer $k$ such that $x = 2k$.

So "Every real number is even" is definitely a (false) statement.

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    $\begingroup$ I didn't downvote, but I feel that that this doesn't really address OP's deeper question: in the usual definition of even, we think of it as applying only to integers. So, under this usual definition, is a statement like "$\pi$ is even" false, or just meaningless? This is actually quite a subtle matter that is handled differently in different foundational systems. $\endgroup$ Commented Jul 28 at 19:24
  • $\begingroup$ @Joe I saw your other comment on my post. I haven't studied the foundations of math yet or what that even means. My only proof-related courses so far are Discrete-math and number-theory. I'm just reading this book to cover any holes I might have in my knowledge. And as my post has shown, I have some holes in my knowledge since I didn't have a answer to my question. Could you attempt to explain to me how my question is handled differently in different foundational systems. $\endgroup$ Commented Jul 28 at 19:59
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    $\begingroup$ @AeroMain27: I'll try to see if can write up an answer in a few days; apologies but I am little busy right now. $\endgroup$ Commented Jul 28 at 20:17
  • $\begingroup$ @Joe no worries. I’ll be looking forward to your answer if you’re able to write one. $\endgroup$ Commented Jul 28 at 21:55
  • $\begingroup$ @AeroMain27: There was an old answer that I wrote to a different question a couple of months ago that was subsequently deleted. I'm trying to get it undeleted – if it is, then I'll send you a link. [Apologies for pinging jigamath again.] $\endgroup$ Commented Jul 30 at 19:45
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Briefly, I'm looking at Hammack's Book of Proof Edition 2.2 (2013). In Section 1.1 he presents the set of even integers as:

$$E = \{2n : n \in \Bbb Z\}$$

He translates this to English as, "E equals the set of all things of form 2n, such that n is an element of $\Bbb Z$".

In the first exercise of Section 2.1, he asks us to categorize the sentence, "Every real number is an even integer". This is a false statement because, for example, the number $1$ is a real number and not twice any integer ($\Bbb Z$).

Note that this is in an elementary context where $\Bbb N \subseteq \Bbb Z \subseteq \Bbb Q \subseteq \Bbb R$, per Example 1.2(7). Being an integer is not actually a prerequisite to be in the set $E$ per the definition (but it's trivially deduced by closure). Hypothetically, if we removed the word "integer" from both the definition and the question here (as the OP wants to do), then it wouldn't change anything.

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    $\begingroup$ The matter is that simple, indeed. Let me iterate my summary comment here: Denote the set of even numbers by $\mathbb{E}$. Given the definition of even numbers by any authoritative reference, there is no category mistake: $\mathbb{E}\subset\mathbb{Z}\subset\mathbb{R}$. Then the statement says $\forall x(x\in\mathbb{R}\rightarrow x\in\mathbb{E})$, which is false. $\endgroup$ Commented Jul 29 at 9:14
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since x is an even integer has a truth value only for integer $\boldsymbol x$′s, the sentence Every real number is an even integer is not a statement. Am I correct?

You're asking whether $\;\forall x \;(x\in \mathbb{R} \implies \exists k{\in}\mathbb Z\;x=2k)\tag*{}$ is a statement. Well, it is a sentence, i.e., a well-formed formula with no free variable, and is a statement in every interpretation (with its attendant axioms/definitions) that admits it.

  1. When working in mathematics with $\mathbb Z$ as the domain of discourse, it isn't really a "statement" (and thus hasn't got a truth value)—just as, in the context of real analysis, it isn't meaningful to work with $\mathbb C$ or its member $3+7i.$
  2. When working in $\mathbb R,$ it certainly is a false statement.

In any case, your premise, "x is an even integer has a truth value only for integer $\boldsymbol x$'s", is false and quite nonsensical: why arbitrarily restrict the context to $\mathbb Z$ when you could just as well restrict it to $\mathbb Z∪\{2.5, 6–9i\}$ instead, or to $\mathbb R$ instead? More plainly: your premise is analogous to claiming, "This is a girl has no truth value when pointing at a table (yet is false when pointing at Tom and true when pointing at Jane)."

My original Question had misquoted the exercise as

...Every real number is even...

and I had kept saying "...is even" whenever I meant "...is an even integer."

In natural language, Colourless green ideas sleep furiously isn't a meaningful sentence/statement, and as such arguably hasn't got a truth value.

  1. If we the word "even" is being defined for integers but not for irrational numbers or imaginary numbers or triangles, then Every real number is even is a statement without a truth value.
  2. Otherwise, Every real number is even and Every good idea is edible are both false statements.

In contrast, to (1) and (2), Ideas green furiously colourless sleep and Is even real every number and bananas are animals and or and 2+3=5 are patently not statements (and thus neither true nor false), because they are syntactically incorrect.

All this was previously discussed at Is $\frac10$ actually false?.

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UPDATED

Depending on the definition used, $\sqrt{2}$ may be either not even, or it may be undefined.

Definition 1

We can define a predicate $Even$ such that:

$\forall x: [x\in N \implies [Even(x)\iff \exists y: [y\in N ~\land ~ x=2\cdot y]]]$

Where $Even(x)$ is taken to meant that $x$ is even.

Since $\sqrt{2}\notin N$, the expression "$Even(\sqrt{2})$", by the above definition, will be undefined. This definition is simply not applicable for $\sqrt{2}$.

Definition 2

Alternatively, we can define the subset of even numbers $E$ such that:

$\forall x: [x \in E \iff a\in N \land \exists y: [y\in N \land y = > 2\cdot x]]$

Then it is trivial to prove: $~~\forall x: [x\notin N \implies x\notin E]$

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    $\begingroup$ The author makes the book freely available online: richardhammack.github.io/BookOfProof . On page 5, the even integers are defined: $E = \{2n : n\in\mathbb{Z}\}$. The misquoted question is on page 38, and should read "Every real number is an even integer." Contrary to the first sentence of your answer, there is no ambiguity. $\endgroup$ Commented Jul 30 at 1:05
  • $\begingroup$ @XanderHenderson If you want to use set-builder notation, more straightforward might be $\{ n\in Z: [\exists x\in Z: n=2x]\}$ corresponding to my second definition. $\endgroup$ Commented Jul 30 at 1:31
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    $\begingroup$ This is not about what I want, it is about what is literally written in the book that this question is about. $\endgroup$ Commented Jul 30 at 1:43
  • $\begingroup$ @XanderHenderson Isn't it obvious that it is false that "Every real number is an even integer?" Is this what we want to formally prove here? We can do this using my 2nd definition and assuming there exists a real number that is not an integer, e.g. $\sqrt{2}$. Am I missing something? $\endgroup$ Commented Jul 30 at 4:41
  • $\begingroup$ The point is that there is context for this question which you have completely ignored. By ignoring that context, you have rather failed to answer the question... You start by saying that "Depending on the definition [of evenness]..." But the definition being considered here is in the book cited in the question. $\endgroup$ Commented Jul 30 at 4:44

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