## lgbasallote 3 years ago 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."

1. joemath314159

So we dont have to worry about negative numbers.

2. lgbasallote

how does that relate?

3. lgbasallote

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

4. wio

What p?

5. farmdawgnation

The p is the conditional.

6. lgbasallote

p succeeds the "if" right?

7. 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.

8. farmdawgnation

That'd be a problem.

9. wio

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

10. joemath314159

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

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

12. lgbasallote

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

13. wio

A if B becomes B only if A

14. wio

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

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

16. wio

necessary but not sufficient basically is the same as necessary.

17. wio

necessary and sufficient is bi-conditional.

18. wio

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

19. lgbasallote

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

20. wio

but clouds don't imply rain.

21. wio

No, not like that.

22. lgbasallote

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

23. 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

24. wio

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

25. 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

26. lgbasallote

so any explanations for that?

27. wio

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

ahh yes. makes sense

29. wio

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

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