Negation of a statement

[Dr.Peterson]

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.

I don't think you've yet shown us the entire exercise as given to you, so we can judge what it is telling you to do. Although ultimately you may just have to ask your instructor, showing us the exact, complete wording may help a lot.

 

[Ming1015]

I don't think you've yet shown us the entire exercise as given to you, so we can judge what it is telling you to do. Although ultimately you may just have to ask your instructor, showing us the exact, complete wording may help a lot.

Opps I'm sorry Dr. Peterson. Well, I only have this exercise which is related to this 'rewrite' form. The complete wording for this exercise is:

A : No prime number is a factor of 11 and 53.

1. Determine the truth value of statement A, then write the negation of statement A.

2. Rewrite statement A in the form ''∀x ∈ D, if P(x), then Q(x)'', where D is a set of all integers. Hence, use this form to write the negation of statement A (in symbolic form and English).

 

[Ming1015]

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.

Yes sir, I'm very clear about this. I just have no idea about the 'rewrite' form in both English and symbolic form, as now the question is asking about writing the statement in ''∀x ∈ D, if P(x), then Q(x)'' where D is a set of all integers.

 

[Dr.Peterson]

Opps I'm sorry Dr. Peterson. Well, I only have this exercise which is related to this 'rewrite' form. The complete wording for this exercise is:

A : No prime number is a factor of 11 and 53.

1. Determine the truth value of statement A, then write the negation of statement A.

2. Rewrite statement A in the form ''∀x ∈ D, if P(x), then Q(x)'', where D is a set of all integers. Hence, use this form to write the negation of statement A (in symbolic form and English).

Thanks. That does confirm that you have accurately quoted the parts you've mentioned. We'll have to wait for your instructor's response (either to a direct question, or to your answers) to be sure exactly what forms are wanted. (If I were helping you in person, I would be looking through your book or notes for hints, but I don't think it's really important.)

 

[pka]

I just have no idea about the 'rewrite' form in both English and symbolic form, as now the question is asking about writing the statement in ''∀x ∈ D, if P(x), then Q(x)'' where D is a set of all integers.

Any \(\mathcal{E}\) proposition , No P is a Q or has the symbolic form \(\left( {\forall x} \right)\left[ {P(x) \Rightarrow \neg Q(x)} \right]\).
The negation is an \(\mathcal{I}\) proposition, Some P is a Q has symbolic form \(\left( {\exists x} \right)\left[ {P(x) \wedge Q(x)} \right]\).
Thus in this case: the \(P(x)\) : x is a prime number and \(Q(x)\) : x is a factor of 11 and 53.

 

[Ming1015]

Thanks. That does confirm that you have accurately quoted the parts you've mentioned. We'll have to wait for your instructor's response (either to a direct question, or to your answers) to be sure exactly what forms are wanted. (If I were helping you in person, I would be looking through your book or notes for hints, but I don't think it's really important.)

Okay, I see. Well, thank you so much for your assist, Dr Peterson. It really helps me a lot.

 

[Ming1015]

Any \(\mathcal{E}\) proposition , No P is a Q or has the symbolic form \(\left( {\forall x} \right)\left[ {P(x) \Rightarrow \neg Q(x)} \right]\).
The negation is an \(\mathcal{I}\) proposition, Some P is a Q has symbolic form \(\left( {\exists x} \right)\left[ {P(x) \wedge Q(x)} \right]\).
Thus in this case: the \(P(x)\) : x is a prime number and \(Q(x)\) : x is a factor of 11 and 53.

Okay, no problem. I think I am clear about it like finally. Thanks for your assist sir!