Is this statement true or false? Find its negation.

logic

Consider the statement
$$\forall x \in \mathbb{R}\;\forall y \in \mathbb{R}\ x+y>0$$
Is this statement true or false? Also, find its negation.

I think this statement is false because the inequality is not always true; for example, for $y=-x$, the inequality is false.

Can you show me how to write the solution in the proper way?

Best Answer

Write:

Since for $x=-1$ and $y=-1$, $(-1)+(-1)=-2>0$ is false. So, the given statement is false.

Clearly, the negation is: $$\exists x,y\in\mathbb{R}\ x+y\leq0$$

DISCUSSION

  1. To show that the statement is false, we just need one counterexample and we are done.

  2. To find the negation, remember that the negative of "for all" is "there exists" and that of $>$ is $\ngtr$ or $\leq$.

Hope this helps. Ask anything if not clear :)

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