Determine the truth value of each statement, assuming that x and y are real numbers, and justify your answer.
- $\forall$x, $\exists$y such that xy=1
- $\exists$y such that $\forall$x , xy=1
I understand the first problem to be true. Since the equation can be "solved" in terms of y, I found that y was a reciprocal of x. So now xy=1 is true. The second problem confuses me. The books says its truth value is false. The only reasoning I have for that answer is the question is the reverse of the first. Can anyone help me understand why the second question is false?