Thread: Determine the truth values of the following statements

1. 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)

2. 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)

Well, we can put them into plain English:

(a) For every integer $a\geq 1$ there is an integer $b\geq 1$ with $ab=\frac{a}{b}$.

This is clearly true by choosing $b=1$ for any $a$.

(b) For every integer $b\geq 1$, there is an integer $a\geq 1$ with $ab=\frac{a}{b}$.

Now, regardless of $a$, it is clear that $b^2=1$. So if $b=2$ then (b) implies $4=1$, which is a contradiction. So (b) is false.

3. Originally Posted by hatsoff
Well, we can put them into plain English:

(a) For every integer $a\geq 1$ there is an integer $b\geq 1$ with $ab=\frac{a}{b}$.

This is clearly true by choosing $b=1$ for any $a$.

(b) For every integer $b\geq 1$, there is an integer $a\geq 1$ with $ab=\frac{a}{b}$.

Now, regardless of $a$, it is clear that $b^2=1$. So if $b=2$ then (b) implies $4=1$, which is a contradiction. So (b) is false.
Thank you very much. Glad to get that one out of the way.