22 Replies  1967 Views  Last Post: 24 June 2013  01:12 PM
#1
Counter definition of prime number?
Posted 24 June 2013  09:49 AM
P(x) ↔ ((x>1) Λ (∀y: D(y,x) → (y=1 ∨ y=x)))
This is my interpretation, is it correct?
P(x) ↔ ((x>1) Λ (¬∃y: D(y,x) → (¬(y=1) ∧ ¬(y=x))))
Replies To: Counter definition of prime number?
#2
Re: Counter definition of prime number?
Posted 24 June 2013  09:53 AM
Let me put it in English and you can see what you think:
X is prime iff x is positive and (if there exists no y which divides x evenly then y is not 1 and y is not x)
This post has been edited by jon.kiparsky: 24 June 2013  09:57 AM
#3
Re: Counter definition of prime number?
Posted 24 June 2013  09:54 AM
You are only inverting the necessary condition, not the sufficient condition, which is only half of the inverse and still incorrect.
Edit Can you write out your definition in plain English? The mathematical representation of your definition is bulky and hard to work with. I'd be interested in hearing your thoughts more plainly.
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P(x) ↔ ((x>1) Λ (¬∃y: D(y,x) → (¬(y=1) ∧ ¬(y=x))))
Edit again I would agree with how Jon read this. I'm just not a fan of it because you're mixing up the necessary and sufficient conditions.
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#4
Re: Counter definition of prime number?
Posted 24 June 2013  10:21 AM
P(x) ↔ ((x>1) Λ (¬∃y: D(y,x) → (¬(y=1) ∧ ¬(y=x))))
According to me it says, x is a prime number iff x is greater than 1 and there does NOT exists a number such that if it divides x then the number does not equal x and 1.
This post has been edited by deprosun: 24 June 2013  10:26 AM
#5
Re: Counter definition of prime number?
Posted 24 June 2013  10:51 AM
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Clearly, let y = 1 or y = x. Thus, your implication does not hold.
Again, order of implication matters. If p > q is a valid implication, then ~q > ~p is a valid implication as well.
When negating a logical statement, if you have a universal quantifier "for all x, p(x)", the negation is "there exists an x such that ~p(x)." Similarly, an existential quantifier is negated by a universal quantifier. So "if there exists an x such that p(x)," the negation is "for all x, ~p(x)."
#6
Re: Counter definition of prime number?
Posted 24 June 2013  11:34 AM
#7
Re: Counter definition of prime number?
Posted 24 June 2013  11:38 AM
#8
Re: Counter definition of prime number?
Posted 24 June 2013  11:39 AM
#9
Re: Counter definition of prime number?
Posted 24 June 2013  12:35 PM
The statement says: A prime number like 7 is relatively prime to any natural number except one of its own multiple.
On a side note: Relatively Prime is when GCD(x,y)=1.
I wanted to translate the bolded statement into predicates as following:
P(x) defines that x is prime
GCD(P(x),y) == 1 → (∀y: ¬D(x,y))
Correct?
#10
Re: Counter definition of prime number?
Posted 24 June 2013  12:41 PM
#11
Re: Counter definition of prime number?
Posted 24 June 2013  12:43 PM
So you're saying my statement is incorrect? How many points will i get out of 10? />
#12
Re: Counter definition of prime number?
Posted 24 June 2013  12:47 PM
#13
Re: Counter definition of prime number?
Posted 24 June 2013  12:48 PM
#14
Re: Counter definition of prime number?
Posted 24 June 2013  12:50 PM
Jimmy Arnold's book is used for the proofs class at my school. Martin Day has a textbook as well, though I prefer Arnold's. Both are free.
http://www.math.vt.e...k/3034Chap1.pdf
#15
Re: Counter definition of prime number?
Posted 24 June 2013  12:54 PM
