Thread: logical statements, does this imply this?

let P(p,q) be a statement about a pair of natural numbers p and q.

does

"there exists a p such that for all q, P(p,q)"

imply

"for all q, there exists a p such that P(p,q)"

i have tried a formal argument but i haven't had much experience with this stuff.

i apologise if this is wrong, as i haven't done discrete math...but doesn't it follow by definition?

eg:

"there exists a p such that for all q, P(p,q). call this P*.

by definition from the previous statement, for all q we have P(p*,q) which is the required result.

that's exactly what i thought, what confused me was that on the worksheet they wanted an example of a statement for which the first was true but not the second. :S

Originally Posted by SpringFan25

"there exists a p such that for all q, P(p,q). call this P*.

by definition from the previous statement, for all q we have P(p*,q) which is the required result.

You are right, though it should be p*, not P*. When one has both p and P, one has to be careful about uppercase and lowercase letters.

So, if one p satisfies P for all q, then for every q we can pick that particular p. Note that the converse is false: if for every q we have its own p that makes P true, in general we can't select a single p that would make P true for all q.

here is the problem (part (b) )

There must be a mistake in part (b) because there is no such property P, as we discussed.