## lgbasallote Group Title 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." one year ago one year ago

1. joemath314159 Group Title

So we dont have to worry about negative numbers.

2. lgbasallote Group Title

how does that relate?

3. lgbasallote Group Title

by p im referring to p and q by the way something like $$p \rightarrow q$$

4. wio Group Title

What p?

5. farmdawgnation Group Title

The p is the conditional.

6. lgbasallote Group Title

p succeeds the "if" right?

7. farmdawgnation Group Title

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.

8. farmdawgnation Group Title

That'd be a problem.

9. wio Group Title

"only if" works like "if" in reverse.

10. joemath314159 Group Title

ah, i misunderstood. I would read that statement as: If a number has no divisors other than one and itself, then it is prime.

11. lgbasallote Group Title

...i didn't know that....

12. lgbasallote Group Title

are there any other phrases that work as if in reverse?

13. wio Group Title

A if B becomes B only if A

14. wio Group Title

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 $$A\rightarrow B$$

15. lgbasallote Group Title

ahh just what i needed. what about necessary but not sufficient?

16. wio Group Title

necessary but not sufficient basically is the same as necessary.

17. wio Group Title

necessary and sufficient is bi-conditional.

18. wio Group Title

The best thing to do is use 'rain' and 'clouds' and see if it makes sense. Then remember that rain implies clouds.

19. lgbasallote Group Title

$(B \rightarrow A) \wedge (\neg A \rightarrow B)$ like that?

20. wio Group Title

but clouds don't imply rain.

21. wio Group Title

No, not like that.

22. lgbasallote Group Title

because i remember something that said necessary but not sufficient and had this solution $(q \rightarrow (\neg r \wedge \neg p)) \wedge \neg((\neg r \wedge \neg p) \rightarrow q)$

23. lgbasallote Group Title

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

24. wio Group Title

I mean you could perhaps say $$(B\rightarrow A) \wedge \neg (A\rightarrow B)$$

25. lgbasallote Group Title

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

26. lgbasallote Group Title

so any explanations for that?

27. wio Group Title

Well if $$B\rightarrow A$$ means necessary and $$A\rightarrow B$$ means sufficient then necessary but not sufficient could be written as $$B\rightarrow A)\wedge \neg (A\rightarrow B$$)

28. lgbasallote Group Title

ahh yes. makes sense

29. wio Group Title

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.

30. lgbasallote Group Title

that's why i hate math.....