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Deduce whether the statement is true or false.

N: If (for any $s,t \in \Bbb R, s^5-st+t^2\geq0$) then (there exsits some $k\in \Bbb R$ such that $k^4+4t^2+5\lt0$).

I'd like to ask isn't it always true that $k^4,t^2$ must be positive for any real $k$? Then $k^4+4t^2+5\lt0$ is false. But why does the question give me $s^5-st+t^2\geq0$? How is this related to the second part of the statement?

Thanks.

sunny
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    You are right in pointing out that the conclusion is false. Note that in an implication i.e. $p \to q$ for statements $p,q$ once $q$ is known to be false, then $p \to q$ , being the same as $(\lnot p) \lor q,$ is the same as simply $\lnot p.$ Thus your answer depends on whether the statement $p$ is true or false. And here $p$ is the statement that for all $s,t$ one has $s^5-st+t^2 \ge 0.$ – coffeemath May 05 '22 at 08:09
  • @coffeemath Ok, I got it! However, I'm not sure how to prove the 'p' statement as it contains 's^5'. How can I guarantee it's positive or negative? May you give me some more hints or any advice? Thank you so much. – sunny May 06 '22 at 06:35
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    In an answer below I point out that you cannot prove the "p" statement because it is false. – coffeemath May 06 '22 at 08:55

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The statement in the question, for which it is asked to determine whether it is true or false, is of the form $$P \implies Q,$$ where $P,Q$ are two (simpler) statements. Thus if it happens that $P$ is false, one can conclude the statement in the question is true, independently of whether or not $Q$ is true. As noted in the OP, it happens here that $Q$ is false. I claim that $P$ is false, so no need to consider $Q$ anyway.

Now $P$ says that, for any two real numbers $s,t,$ one has $$s^5-st+t^2 \ge 0.$$ But then $P$ is false, because taking $s=-1,t=0$ this last is $(-1)^5-(-1)\cdot 0 +0^2 \ge 0,$ which computes to $-1 \ge 0,$ a false statement.

coffeemath
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