Question

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."

Answer 1

So we dont have to worry about negative numbers.

Answer 2

how does that relate?

Answer 3

by p im referring to p and q by the way

something like \(p \rightarrow q\)

Answer 4

What p?

Answer 5

The p is the conditional.

Answer 6

p succeeds the "if" right?

Answer 7

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.

Answer 8

That'd be a problem.

Answer 9

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

Answer 10

ah, i misunderstood. I would read that statement as:

If a number has no divisors other than one and itself, then it is prime.

Answer 11

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

Answer 12

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

Answer 13

A if B becomes B only if A

Answer 14

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 \(A\rightarrow B\)

Answer 15

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

Answer 16

necessary but not sufficient basically is the same as necessary.

Answer 17

necessary and sufficient is bi-conditional.

Answer 18

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

Answer 19

\[(B \rightarrow A) \wedge (\neg A \rightarrow B)\]
like that?

Answer 20

but clouds don't imply rain.

Answer 21

No, not like that.

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

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

Answer 26

so any explanations for that?

Answer 27

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

Answer 28

ahh yes. makes sense

Answer 29

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.

Answer 30

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