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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.

Can someone please confirm my answer and help me explain it? I will gladly appreciate it!

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    $\begingroup$ 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. $\endgroup$
    – lulu
    Commented Nov 18, 2020 at 11:08
  • $\begingroup$ @lulu Oh sorry, that was a typo. I will correct it. Thank you for letting me know! $\endgroup$
    – Andy
    Commented Nov 18, 2020 at 11:59

1 Answer 1

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The expression to hold is $$x+y<0$$

Now let $x=-y-k$ expression becomes $$(-y-k)+y<0 \Leftrightarrow$$ $$/\text{ Associative rule} : -y-k=-k-y/$$ $$-k-y+y<0\Leftrightarrow$$ $$-k<0\Leftrightarrow$$ $$0<k$$

Now we can select any $k>0$ and the proposition will hold.

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