- Thread starter
- #1

#### skatenerd

##### Active member

- Oct 3, 2012

- 114

True or False. Assume standard properties of real numbers. Support your answers.

(a) For all \(x\) in the real numbers, there exists a \(y\) in the real numbers such that \(x+y>0\).

The answer: True. Given an \(x\), take \(y=|x|+1\).

(b) There exists some \(x\) in the real numbers such that for all \(y\) in the real numbers, \(x+y>0\).

The answer: False. There is no single \(x\) that can "accommodate" all \(y\)'s.

I understand how the first answer makes sense, because taking \(y=|x|+1\) would give you \(x+|x|+1>0\). For any \(x<0\), we would have \(1>0\). Any \(x>0\) would clearly give something more than 1.

However when looking at part (b), I feel like it is exactly the same thing as part (a), except the variable \(x\) is now the constant and the constant \(y\) is now the variable. So why would it not be valid to say:

"True: Given a \(y\), take \(x=|y|+1\)"?