Discrete math logic question

Question

I have the following two questions.

1. For all real numbers x, there is a real number y such that $2x+y=7$ would this be true or false?

I think true because if you put

$2(7)+y=14$

$2(8)+y=14$ there will always be a specific y that will make it work is this logic correct.

1. There is a real numbers x that for all real number y, $2x+y=7$ will be true.

would this be false because if you say $x=6$

then you get

$2(6)+2=14$

only if $y=2$ would it work but it would not work for every y.

Answer 1

Yes, indeed, you are correct in your assessment of the truth or falsity of each statement.

In the first, we can see this as allowing $y$ to depend on $x$. So for any given $x$, we can find some $y$, and in particular, we can simply choose $y = 7 - 2x$ which will guarantee the equalition holds.

In the second case, $y$ cannot depend on any given $x$. For the statement to be true, we need to consider the existence of a particular $y$ such that for every $x$, regardless of what $x$ may be, the equality holds. Since $x$ can vary, but $y$ can not vary accordingly, the statement is clearly false.

These two statements help demonstrate just how crucial the order of quantifiers and quantified variables can be: in the first, we have a true statement, and in the second, a false statement, and the only difference between them is the placement of $\exists y \cdots$.

Answer 2

Yes, you're on the right track. Both statements can be more easily analyzed by rearranging to get

$$x = \frac{14 - y}{2}$$

This should allow you to fill in the gaps in your justification on (1) (i.e. not just stating a few examples in which it works).