# For every integer $y$, there is an integer $x$ such that $x + y < 0$

Determine the truth value of the following statement and explain your answer.

Statement: For every integer $$y$$, there is an integer $$x$$ such that $$x + y < 0$$.

I think the truth value is true but I'm not sure how to explain why since we're not allowed to use examples in our explanation. I tried converting the statement into the following formula

$$\forall y \exists x \left( x + y < 0 \right)$$

but don't quite know where to go from there.

• Your formal version of the claim introduces a "$10$" where it ought to have a "$0$" but is otherwise ok. For the claim itself, taking $x=-1-y$ suffices.
– lulu
Commented Nov 18, 2020 at 11:08
• @lulu Oh sorry, that was a typo. I will correct it. Thank you for letting me know!
– Andy
Commented Nov 18, 2020 at 11:59

The expression to hold is $$x+y<0$$
Now let $$x=-y-k$$ expression becomes $$(-y-k)+y<0 \Leftrightarrow$$ $$/\text{ Associative rule} : -y-k=-k-y/$$ $$-k-y+y<0\Leftrightarrow$$ $$-k<0\Leftrightarrow$$ $$0
Now we can select any $$k>0$$ and the proposition will hold.