I am just being introduced to quantifiers in logic and my lecturer was going through the following two statements. The question is to determine which, if any, is/are true.
- $(\forall x \in \mathbb{R})(\exists y \in \mathbb{R})[x + y = 0]$
- $(\exists x \in \mathbb{R})(\forall y \in \mathbb{R})[x + y = 0]$
Clearly, the first statement is true; we can just let $y = -x$. However, my lecturer says that the second statement is false. I cannot wrap my head around why that is the case. If we can take $y = -x$ in the first, why can we not do the same for the second i.e. let $x = -y$? In fact, how is the second statement any different from the first?
Any intuitive explanations/examples would be greatly appreciated!
Best Answer
The second one is false because there is no real number $x$ such that its addition with all real numbers results in $0$. You can take $y=x^2+1$ for each $x$ and observe that $$x+y=x^2+x+1\ne 0$$