Thread: Determine the truth values of the following statements

Let S(a,b) be the statement "ab=a/b".

Given that the domain (or universe of discourse) for both variables consists of all positive, non-zero integers, what are the truth values of these two statements?

a) ∀a∃b S(a,b)
b) ∀b∃a S(a,b)

Explain your reasoning when you answer.

Originally Posted by Runty

Let S(a,b) be the statement "ab=a/b".

Given that the domain (or universe of discourse) for both variables consists of all positive, non-zero integers, what are the truth values of these two statements?

a) ∀a∃b S(a,b)
b) ∀b∃a S(a,b)

Explain your reasoning when you answer.

Well, we can put them into plain English:

(a) For every integer there is an integer with .

This is clearly true by choosing for any .

(b) For every integer , there is an integer with .

Now, regardless of , it is clear that . So if then (b) implies , which is a contradiction. So (b) is false.

Originally Posted by hatsoff

Well, we can put them into plain English:

(a) For every integer there is an integer with .

This is clearly true by choosing for any .

(b) For every integer , there is an integer with .

Now, regardless of , it is clear that . So if then (b) implies , which is a contradiction. So (b) is false.

Thank you very much. Glad to get that one out of the way.