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So we dont have to worry about negative numbers.
how does that relate?
by p im referring to p and q by the way something like \(p \rightarrow q\)
The p is the conditional.
p succeeds the "if" right?
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.
That'd be a problem.
"only if" works like "if" in reverse.
ah, i misunderstood. I would read that statement as: If a number has no divisors other than one and itself, then it is prime.
...i didn't know that....
are there any other phrases that work as if in reverse?
A if B becomes B only if A
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 \(A\rightarrow B\)
ahh just what i needed. what about necessary but not sufficient?
necessary but not sufficient basically is the same as necessary.
necessary and sufficient is bi-conditional.
The best thing to do is use 'rain' and 'clouds' and see if it makes sense. Then remember that rain implies clouds.
\[(B \rightarrow A) \wedge (\neg A \rightarrow B)\] like that?
but clouds don't imply rain.
No, not like that.
because i remember something that said necessary but not sufficient and had this solution \[(q \rightarrow (\neg r \wedge \neg p)) \wedge \neg((\neg r \wedge \neg p) \rightarrow q)\]
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
I mean you could perhaps say \((B\rightarrow A) \wedge \neg (A\rightarrow B)\)
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
so any explanations for that?
Well if \(B\rightarrow A\) means necessary and \(A\rightarrow B\) means sufficient then necessary but not sufficient could be written as \(B\rightarrow A)\wedge \neg (A\rightarrow B\))
ahh yes. makes sense
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.
that's why i hate math.....