Negation of "either $x = 0$ or $y = 0$"

Question

Why is the negation of "either $x = 0$ or $y = 0$" both $x \neq 0$ and $y \neq 0$?

Or is inclusive here, I suppose?

An unqualified "or" is always inclusive in mathematics and computer science. If exclusive or is intended, then it will always be specified explicitly. — Matt Samuel, Aug 31 '16 at 18:46
@quid I've never encountered a case where "either/or" was used in mathematics/computer science and inclusive or was not intended. I obviously haven't read every single book and paper, but I would be very confused if someone used "either/or" in a math paper and intended exclusive or without specifying "and not both." — Matt Samuel, Aug 31 '16 at 18:51
Reading the first question as stated, one could answer "because the or is inclusive". — , Aug 31 '16 at 18:53
@MattSamuel yes, but maybe only because those that know it actually is exclusive usually also know that not everybody knows this and want to be on the safe side ;-) // More to the point in any case I strongly believe that the motivation for the questions was the "either" and to drop it in an answer is thus confusing. — quid, Aug 31 '16 at 18:54
Actually, the next problem in my book cleared this up for me I think. We have to negate "the integers a and b are both even", and the answer is "either the integer a is odd or the integer b is odd". So to make the first statement false, it sufficient that only one of a or be is not even (odd). Is this correct thinking? — Max, Aug 31 '16 at 19:10
@Max this example indeed makes clear that this text uses "either...or..." to mean inclusive or, so that "both" is a valid answer. I think this is poor style though as the "either" then serves no purpose yet might create confusion. Since "either...or..." is considered to be an exclusive or by some; that is the presence of "either" indicates it is exclusive. — quid, Aug 31 '16 at 19:14
@quid You are needlessly confusing the OP. "or" is inclusive whether "either" is used or not. The usage in spoken English has no bearing on the usage in mathematics. — Matt Samuel, Aug 31 '16 at 19:28
@MattSamuel the discussion I linked to says something else. The context there is mathematics. — quid, Aug 31 '16 at 19:40
The accepted answer seems to be speaking of its use in natural language. You don't write "select either a or b" in a proof. Even if you did, if it's known that you're only selecting one, it doesn't matter whether it's inclusive or exclusive, the meaning is the same. If you're possibly selecting more than one, it would default to inclusive or. — Matt Samuel, Aug 31 '16 at 19:49
@MattSamuel pleas explain why when used in a mathematical text the word "either" should become a meaningless filler word while in standard English 'The Oxford English Dictionary explains "either ... or" as follows: "The primary function of either, etc., is to emphasize the perfect indifference of the two (or more) things or courses ... ; but a secondary function is to emphasize the mutual exclusiveness, = either of the two, but not both."" (from Wikipedia page exclusive or) I know that it does everywhere including maths, but it's sloppy. — quid, Aug 31 '16 at 19:57
@quid "either" would be used to prevent the exposition from being dreary, not to indicate exclusive or. Meaningless fillers are not strange, and even spoken English has them ("do" is often meaningless but required). The Oxford English Dictionary is a world-renowned reference for the English language, but even the best general language dictionary has sparse coverage of the usage of words in mathematics or other technical specialties at best. Their primary function is to define words in spoken or non-technical written English. — Matt Samuel, Aug 31 '16 at 20:02
@MattSamuel And I think it's a poor decision to use in this way, as it can lead to confusion. — quid, Aug 31 '16 at 20:33
@quid Then I guess this is the point where we agree to disagree! — Matt Samuel, Aug 31 '16 at 20:36
By they way, what does "negate" mean precisely (although I doubt it's defined in any way)? Does it mean we need a statement that would be true for all false cases, and false for every true cases of the original statement? — Max, Aug 31 '16 at 20:46
@Max Yes, that's how the negation of a statement is defined. — Matt Samuel, Aug 31 '16 at 20:49
@MattSamuel Alright. Let me add that I do not mean this dogmatically, say, if somebody wrote: "we can now conclude that A is compact either by observing that it is the image of the compact set B under the continuous map f or by checking directly that it is bounded and closed" then I would not object on the grounds that one could in principle also do both. Or something like "thus the group G must be either simple or abelian" and things like this. — quid, Aug 31 '16 at 22:34
Yet to go back to the start, I believe that the usage in the question is poor style and also not all that common, as I think people tend to use "either" more in context where they want to stress if not strict exclusiveness then still a certain degree of independence or orthogonality. Say, "let $n$ be even and $n= ab$, then either $a$ is even or $b$ is even" is not good in my opinion. — quid, Aug 31 '16 at 22:34
@quid The answerers of this question didn't even think to interpret it exclusively until you said something. It's extremely common to phrase it this way. I don't see any problem with it. In English, simply using "or" even without "either" implies that it's exclusive. If you object to "either", you should object to using "or" this way as well. — Matt Samuel, Aug 31 '16 at 22:41
@MattSamuel Yes I even commented that on an answer giving some "intuitive" real word example. And "either" stresses the exclusiveness. Therefore it is surely a great idea from a pedagogical point of view to use an "either" there that serves no purpose :-/ Note, the question also asked if the "or" is inclusive. How can you claim it did not occur to asker it might be exclusive is beyond me. I think they were (rightly) incredulous it could be inclusive. But yes, from context it's clear it was really meant like this. Poor style. You can have the last word if you like. — quid, Aug 31 '16 at 23:04
I tend to read "either $a$ is even or $b$ is even" as a shorter form of "either $a$ is even or else $b$ is even", that is, "if $a$ is not even then $b$ is even". The inclusiveness of the "either/or" matches the possibility of vacuous implication in the "if/then" — Carl Mummert, Aug 31 '16 at 23:31

score 5 · Accepted Answer · answered Aug 31 '16 at 18:48

5

It is inclusive, unless specified otherwise.

Let $x=0$ be $A$ and $y=0$ be $B$.

By De Morgan's Law $$ \neg (A \lor B) = (\neg A \land \neg B) $$

However, it does say either. This might indicate it is exclusive. In that case, that negation is not correct.

answered Aug 31 '16 at 18:48

Samasambo

351

1

Looks like it's inclusive. The answer is given by the book. – Max Aug 31 '16 at 19:11

score 3 · Answer 2 · answered Aug 31 '16 at 18:47

3

Yes it is inclusive if it is not precised.

In general the negation of "$A$ or $B$" is "not $A$ and not $B$". To convince yourself that it is true, you can compare the truth tables.

answered Aug 31 '16 at 18:47

C. Dubussy

9,120

but it says "either...or..." which I think is the point of the question. – quid Aug 31 '16 at 18:49
So is this all that matters in mathematics, or should I be able to convince myself intuitively in that? – Max Aug 31 '16 at 19:05
Of course you can (and you should) also be convinced intuitively. Try for example to negate the sentence "I'll eat an apple or an orange". – C. Dubussy Aug 31 '16 at 19:13
@C.Dubussy honestly that type of example add to the confusion since in everyday language "or" in such an everyday context is often intended as an exclusive or. – quid Aug 31 '16 at 19:19

score 0 · Answer 3 · answered Sep 28 '16 at 09:36

statement A: either x=0 or y=0. negation of statement A: Its not true that either x=0 or y=0. as negation means opposite of statement A , so we are saying that statement-A is not true. now you read negation statement once it says that both are false it is similar to I wont eat either pasta or french fries which is equivalent to say that I wont eat pasta and I wont eat french fries. So if you equate you will get as Its not true that x=0 and its not true that y=0.

to write in short: ~(p or q) = ~p And ~q ~(p and q) = ~p or ~q if you know about truth tables you can even compare them with truth tables. if anyone needs detailed proof using truth tables do ask :-)

Negation of "either $x = 0$ or $y = 0$"

3 Answers3