lgbasallote
  • lgbasallote
Logic Question: A positive integer is prime only if it has no other divisors other than 1 and itself. Why is the p here "A positive integer is prime" and not "it has no other divisors other than 1 and itself."
Mathematics
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katieb
  • katieb
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anonymous
  • anonymous
So we dont have to worry about negative numbers.
lgbasallote
  • lgbasallote
how does that relate?
lgbasallote
  • lgbasallote
by p im referring to p and q by the way something like \(p \rightarrow q\)

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anonymous
  • anonymous
What p?
farmdawgnation
  • farmdawgnation
The p is the conditional.
lgbasallote
  • lgbasallote
p succeeds the "if" right?
farmdawgnation
  • farmdawgnation
So you'd rewrite this as, If positive integer n is prime, it has no other divisors other than 1 and itself. However, you CAN'T say.. If n has no other divisors than 1 and itself, it is a positive integer.
farmdawgnation
  • farmdawgnation
That'd be a problem.
anonymous
  • anonymous
"only if" works like "if" in reverse.
anonymous
  • anonymous
ah, i misunderstood. I would read that statement as: If a number has no divisors other than one and itself, then it is prime.
lgbasallote
  • lgbasallote
...i didn't know that....
lgbasallote
  • lgbasallote
are there any other phrases that work as if in reverse?
anonymous
  • anonymous
A if B becomes B only if A
anonymous
  • anonymous
A is necessary for B means \(B\rightarrow A\). A is sufficient for B means \(A\rightarrow B\). A if B means \(B\rightarrow A\) A only if B means \(A\rightarrow B\)
lgbasallote
  • lgbasallote
ahh just what i needed. what about necessary but not sufficient?
anonymous
  • anonymous
necessary but not sufficient basically is the same as necessary.
anonymous
  • anonymous
necessary and sufficient is bi-conditional.
anonymous
  • anonymous
The best thing to do is use 'rain' and 'clouds' and see if it makes sense. Then remember that rain implies clouds.
lgbasallote
  • lgbasallote
\[(B \rightarrow A) \wedge (\neg A \rightarrow B)\] like that?
anonymous
  • anonymous
but clouds don't imply rain.
anonymous
  • anonymous
No, not like that.
lgbasallote
  • lgbasallote
because i remember something that said necessary but not sufficient and had this solution \[(q \rightarrow (\neg r \wedge \neg p)) \wedge \neg((\neg r \wedge \neg p) \rightarrow q)\]
lgbasallote
  • lgbasallote
the statement was this: For hiking to be safe, it is necessary but not sufficient that berries be not ripe along the trail and for grizzly bears not to have been seen in the area
anonymous
  • anonymous
I mean you could perhaps say \((B\rightarrow A) \wedge \neg (A\rightarrow B)\)
lgbasallote
  • lgbasallote
the p, q, r are these: p: Grizzly bears have been sean in the area q: hiking is safe r: berries are ripe along the trail
lgbasallote
  • lgbasallote
so any explanations for that?
anonymous
  • anonymous
Well if \(B\rightarrow A\) means necessary and \(A\rightarrow B\) means sufficient then necessary but not sufficient could be written as \(B\rightarrow A)\wedge \neg (A\rightarrow B\))
lgbasallote
  • lgbasallote
ahh yes. makes sense
anonymous
  • anonymous
But there is a big of an ambiguity for me, because when they say necessary and not sufficient, do they mean necessary but not necessarily sufficient, or necessary and never sufficient.
lgbasallote
  • lgbasallote
that's why i hate math.....

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