Negation of a statement

[Ming1015]

The statement is 'No prime number is a factor of 11 and 53'. What is the negation of this statement?

 

[Dr.Peterson]

The negation is "It is not true that no prime number is a factor of 11 and 53".

You are probably expected to restate that more directly.

What have you learned about negating a statement of the form "No X is Y"? Are you aware that the negation has the form "Some X is Y"?

 

[Ming1015]

The negation is "It is not true that no prime number is a factor of 11 and 53".

You are probably expected to restate that more directly.

What have you learned about negating a statement of the form "No X is Y"? Are you aware that the negation has the form "Some X is Y"?

Yes, I learnt that form of negation. Should I answer it as 'Some prime numbers are the factors of 11 and 53' or I need to change the term 'and' to the term 'or' in the sentence? Because I'm not very sure about it but I found out 'Some prime numbers are the factors of 11 or 53' seems more correctly for me

 

[Dr.Peterson]

Yes, I learnt that form of negation. Should I answer it as 'Some prime numbers are the factors of 11 and 53' or I need to change the term 'and' to the term 'or' in the sentence? Because I'm not very sure about it but I found out 'Some prime numbers are the factors of 11 or 53' seems more correctly for me

Thanks for clarifying your question. I think it is a matter of the subtlety of English.

The original statement, "No prime number is a factor of 11 and 53", might more clearly have been written as "No prime number is a factor of both 11 and 53 ", or "No prime number is a common factor of 11 and 53". It means "There exists no prime number that is a factor of both 11 and 53".

Its negation would be, "There exists a prime number that is a factor of 11 and 53" (where all I did was to remove the word "no"), or "Some prime number is a factor of both 11 and 53".

Your negation, "Some prime numbers are the factors of 11 and 53", is almost right, but the word "the" changes its meaning. The trouble is that "the factors" implies a complete list of factors (which is meaningless when two numbers are given!); that in turn suggests that, instead, 11 and 53 are the factors, changing it to a special use of "of" that is not what you intend here. (I also prefer the singular form, as I stated it, which I think is clearer, but is not technically necessary.)

You ask about a different issue, namely whether the conjunction should be changed to "or". I suppose you think this because of De Morgan's law, in which negation makes such a change. But you have not negated (and should not negate) the phrase "a factor of 11 and 53"; you would only do that if you wanted to say "There is a prime number that is not a factor of 11 and 53", which could be written as "There is a prime number that is not a factor of 11 or is not a factor of 53". But that is not where you need to negate.

This sort of language difficulty is why we prefer to use symbols; they are much easier to be sure of.

 

[Ming1015]

I see, thanks for your reply. So it means that actually we don't really need to negate the 'and' at the behind of the sentence for this type of sentence ? I'm sorry as I was a little confused about it. Then, does it means that I can write the negation as 'There exists a prime number that is a factor of 11 and 53'? Anyway, should I consider the original statement is true? The reason why I said it was true is because that the both numbers 11 and 53 are already prime numbers and they are a factor of themselves respectively.

 

[Dr.Peterson]

I see, thanks for your reply. So it means that actually we don't really need to negate the 'and' at the behind of the sentence for this type of sentence ? I'm sorry as I was a little confused about it. Then, does it means that I can write the negation as 'There exists a prime number that is a factor of 11 and 53'?

Yes; and I think the fact that all I did was to remove the word "no" from my rewritten form of the original makes that clear.

Anyway, should I consider the original statement is true? The reason why I said it was true is because that the both numbers 11 and 53 are already prime numbers and they are a factor of themselves respectively.

The original statement, "No prime number is a factor of 11 and 53", is clearly true; the only (common) factor of 11 and 53 is 1, which is not prime. I'm not sure how "they are a factor of themselves respectively" is relevant, unless what you mean is that each is the only factor of itself other than 1.

 

[Steven G]

Dr Peterson, I was wondering if you would accept the negation of P as it is not the case that P and nothing else?

For this post the negation is it is not the case that No prime number is a factor of 11 and 53.

 

[Dr.Peterson]

Dr Peterson, I was wondering if you would accept the negation of P as it is not the case that P and nothing else?

For this post the negation is it is not the case that No prime number is a factor of 11 and 53.

Yes, that's the first thing I said, with only a slight difference of wording:

The negation is "It is not true that no prime number is a factor of 11 and 53".

You are probably expected to restate that more directly.

What have you learned about negating a statement of the form "No X is Y"? Are you aware that the negation has the form "Some X is Y"?

If I put a question like this on a test, I would probably want to find a way to prevent the simple answer, but otherwise it would be acceptable.

 

[Ming1015]

[MATH] [QUOTE="Dr.Peterson, post: 514598, member: 62318"] Yes; and I think the fact that all I did was to remove the word "no" from my rewritten form of the original makes that clear. The original statement, "No prime number is a factor of 11 and 53", is clearly true; the only (common) factor of 11 and 53 is 1, which is not prime. I'm not sure how "they are a factor of themselves respectively" is relevant, unless what you mean is that each is the [I]only [/I]factor of itself other than 1. [/QUOTE] Sorry for my confusing reply. 'They are a factor of themselves' I meant is 11 and 53 are already a factor of both 11 and 53 respectively, and both 11 and 53 are already prime numbers. So I thought the [B]original[/B] statement maybe will be [B]false[/B] as there is a prime number that is a factor for both these numbers, but it's the case that the factor is [B]not common[/B]. However, I reconsidered about this statement, and found out it to be clearly true though as the only (common) factor is 1, which is not a prime number, which proves the statement is true. Anyway thanks for your clear explanation, Dr. Peterson. I would like to ask about one more question again. If I am asked to [B]rewrite the original statement in the form ''∀x ∈ D, if P(x), then Q(x)'' where D is a set of all integers[/B], how should I rewrite it correctly?? [/MATH]

 

[Cubist]

If I am asked to rewrite the original statement in the form ''∀x ∈ D, if P(x), then Q(x)'' where D is a set of all integers, how should I rewrite it correctly??

Have you got any thoughts about this? Perhaps see if you can make this statement equivalent to your original:-
Given x is any integer, if <something about x is true> then <something else about x is true>

 

[Ming1015]

Have you got any thoughts about this? Perhaps see if you can make this statement equivalent to your original:-
Given x is any integer, if <something about x is true> then <something else about x is true>

For me, I will write it as 'Given that x is an integer, if x is a prime number, then x is not a factor of 11 and 53.' but of course, I'm not sure about my answer is correct or not and I'm not sure whether it's necessary to state which is P(x) and which is Q(x).

 

[Cubist]

For me, I will write it as 'Given that x is an integer, if x is a prime number, then x is not a factor of 11 and 53.' but of course, I'm not sure about my answer is correct or not and I'm not sure whether it's necessary to state which is P(x) and which is Q(x).

I agree with with your statement. So that's probably good (unless I'm wrong too)!

Obviously P(x) would correspond to "x is a prime number" and Q(x) corresponds to "x is not a factor of 11 and 53". However I'm not sure how you'd put this into proper mathematical notation myself. For that you'll have to wait for @Dr.Peterson or another helper. Do you have any notes regarding the required notation? If so then I recommend that you give it a try yourself and post back (so that others can check it)

 

[pka]

The statement is 'No prime number is a factor of 11 and 53'. What is the negation of this statement?

This link does a fairly good job of explaining the square of opposition .
Across the square the negation of no P is Q is some P is not Q. Across the square the negation of no P is Q is some P is Q. ...... [edited]
Thus the negation of 'No prime number is a factor of 11 and 53' is some prime number is a factor of 11 and 53.
The textbook Symbolic Logic by Copi is better.

 

[Dr.Peterson]

For me, I will write it as 'Given that x is an integer, if x is a prime number, then x is not a factor of 11 and 53.' but of course, I'm not sure about my answer is correct or not and I'm not sure whether it's necessary to state which is P(x) and which is Q(x).

I would expect the answer to include a definition of P and Q, and to express the statement symbolically:

Let P(x) = "x is a prime number" and Q(x) = "x is not a factor of both 11 and 53". Then ∀x ∈ Z, if P(x), then Q(x).

You may be allowed to replace Q(x) with a compound statement. That depends on how the problem was actually worded, and what examples you have seen of this sort.

 

[pka]

This link does a fairly good job of explaining the square of opposition .
Across the square the negation of no P is Q is some P is not Q.
Thus the negation of 'No prime number is a factor of 11 and 53' is some prime number is a factor of 11 and 53.
The textbook Symbolic Logic by Copi is better.

EDIT CORRECTION: Across the square the negation of no P is Q is some P is Q. ........ [response #13 edited with suggested correct version]

 

[Ming1015]

[MATH] [QUOTE="Cubist, post: 514643, member: 75980"] I agree with with your statement. So that's probably good (unless I'm wrong too)! Obviously P(x) would correspond to "x is a prime number" and Q(x) corresponds to "x is not a factor of 11 and 53". However I'm not sure how you'd put this into proper mathematical notation myself. For that you'll have to wait for [USER=62318]@Dr.Peterson[/USER] or another helper. Do you have any notes regarding the required notation? If so then I recommend that you give it a try yourself and post back (so that others can check it) [/QUOTE] Okay I see, thanks for the assist too. Unfortunately I'm still a student and new to this syllabus, I don't have much notes for that. Okay maybe I will try my best to complete it first. [/MATH]

Let P(x) = "x is a prime number" and Q(x) = "x is not a factor of both 11 and 53". Then ∀x ∈ Z, if P(x), then Q(x).

You may be allowed to replace Q(x) with a compound statement. That depends on how the problem was actually worded, and what examples you have seen of this sort.

Thanks Dr. Peterson for your explanation. There are still two questions pop out in my mind:
1. Does it mean that I have to changed the D I mentioned into Z in ∀x ∈ D? If I'm not mistaken, Z is also a set of all integers which same as the D as I mentioned in the question I sent, or the Z mentioned is just same as D only? I'm sorry but just wanna confirm bout it.
2. Does it mean that the negative statement (statement with 'not') actually can be directly taken as a predicate Q(x) without modifying it to positive? Because as I know, for the statements, we usually take the positive one to be 'let'. I don't know if it is the same goes to the predicates.

 

[Ming1015]

This link does a fairly good job of explaining the square of opposition .
Across the square the negation of no P is Q is some P is not Q. Across the square the negation of no P is Q is some P is Q. ...... [edited]
Thus the negation of 'No prime number is a factor of 11 and 53' is some prime number is a factor of 11 and 53.
The textbook Symbolic Logic by Copi is better.

Thanks for your information sharing, I looked through the link already and it's quite helpful for me

 

[Dr.Peterson]

Okay I see, thanks for the assist too. Unfortunately I'm still a student and new to this syllabus, I don't have much notes for that. Okay maybe I will try my best to complete it first.

Thanks Dr. Peterson for your explanation. There are still two questions pop out in my mind:
1. Does it mean that I have to changed the D I mentioned into Z in ∀x ∈ D? If I'm not mistaken, Z is also a set of all integers which same as the D as I mentioned in the question I sent, or the Z mentioned is just same as D only? I'm sorry but just wanna confirm bout it.
2. Does it mean that the negative statement (statement with 'not') actually can be directly taken as a predicate Q(x) without modifying it to positive? Because as I know, for the statements, we usually take the positive one to be 'let'. I don't know if it is the same goes to the predicates.

1. My understanding of 'rewrite the original statement in the form ''∀x ∈ D, if P(x), then Q(x)'' where D is a set of all integers' would be that the names D, P, and Q can be replaced with other names; but you could either use Z, which is already defined (as the set of all integers, not a set of all integers!), or define D as such. But the set you use has to be defined in order to use it. But this is the sort of thing you ask your teacher, or get from examples you have been shown. It's partly a matter of "taste".

2. You can define a predicate any way you want; you can also use all sorts of statements following "∀x ∈ ...". Again, you should be able to look at examples in your book to see some of the variety possible (though you may be early enough in your course that all examples so far are very simple). So both what I said and

"Let P(x) = "x is a prime number" and Q(x) = "x is a factor of both 11 and 53". Then ∀x ∈ Z, if P(x), then ~Q(x).​

can be valid (though your teacher may have a preference).

 

[Ming1015]

[MATH][/MATH]

1. My understanding of 'rewrite the original statement in the form ''∀x ∈ D, if P(x), then Q(x)'' where D is a set of all integers' would be that the names D, P, and Q can be replaced with other names; but you could either use Z, which is already defined (as the set of all integers, not a set of all integers!), or define D as such. But the set you use has to be defined in order to use it. But this is the sort of thing you ask your teacher, or get from examples you have been shown. It's partly a matter of "taste".

2. You can define a predicate any way you want; you can also use all sorts of statements following "∀x ∈ ...". Again, you should be able to look at examples in your book to see some of the variety possible (though you may be early enough in your course that all examples so far are very simple). So both what I said and

"Let P(x) = "x is a prime number" and Q(x) = "x is a factor of both 11 and 53". Then ∀x ∈ Z, if P(x), then ~Q(x).​

can be valid (though your teacher may have a preference).

Okay I see... seems like I'm more clear about it now. Anyway, how should I write this statement in both symbolic form and English?

In my opinion,
Let P(x) = "x is a prime number",
Let Q(x) = "x is not a factor of both 11 and 53".

Symbolic form: ∀x ∈ D, P(x)→Q(x).

English: For each integer x, if x is a prime number, then x is not a factor of 11 and 53.

Well, this is just only my answer. I'm not sure whether it is appropriate or not, especially the symbolic form.

 

[pka]

English: For each integer x, if x is a prime number, then x is not a factor of 11 and 53.

In any standard textbook on Symbolic Logic we would find that the negation of an \(\mathcal{E}\) proposition
"No P is Q" is an \(\mathcal{I}\) proposition. "
Some P is a Q." Thus Some prime number is a factor of 11 and 53, is the negation of No prime number is a factor of 11 and 53.