0
$\begingroup$

So I want to just check if my my critical reading of these problems are correct. I want to see if my answers match up.

(a) $\forall x\in\mathbb{Z}$: if $x^2\leqslant 30$, then $x\leqslant5$.

(b) $\forall x\in\mathbb{R}$: if $x^2\leqslant 30$, then $x\leqslant 5$.

(c) $\forall x\in\mathbb{Z}$: if $x\leqslant 5$, then $x^2\leqslant 30$.

(d) $\forall x\in\mathbb{R}_+$: if $x\leqslant 5$, then $x^2\leqslant 30$.

It seems like the first three statements are false and the last one is true but I could be mistaken. Thoughts?

$\endgroup$
  • $\begingroup$ a) is not correct $-5\leq x\leq 5$, b) $\sqrt{-5}\leq x\leq\sqrt{5}$ c) what about $x=-8$ d) is true since $x^2$ is an increasing function on $[0, \infty)$ thus for $0\le x\leq 5$ we have $0\leq x^2\leq 25\leq 30$ $\endgroup$ – R.N Nov 11 '15 at 22:36
0
$\begingroup$

Looks like you are right.

If $x\in\mathbb{Z}$, that implies $x$ is an integer. If $x\in\mathbb{R}$, that implies $x$ is a real number. Know the difference.

Looking at (a), if $x \le 30$, and $x\in\mathbb{Z}$, then $-5 \le x \le 5$, since $6^2 = 36 \ge 30$. Thus, (a) is false.

Note the slight difference in part (b). $x \le 30$, but this time $x\in\mathbb{R}$. Since $x$ can be any real number, $x \ge 5$. In fact, the inequality follows that $-\sqrt{30} \le x \le \sqrt{30}$. Thus, (b) is false.

Looking at (c), if $x \le 5$, then $x^2 \le 25$. However, since $x\in\mathbb{Z}$, $x^2$ could be greater than $30$ (i.e. $x = -6$). Therefore, part (c) is false.

Looking at (d), if $x \le 5$, then $x^2 \le 25$. Therefore, $x \le 30$. This is similar to the question in part (c).

Hope this helped. Comment if you have questions.

$\endgroup$
  • $\begingroup$ I'm confused as why you concluded (a) is false. Surely $-5\le x \le 5$ implies $x\le 5$. See the Comment by @Takhteh_pareh about this a few seconds before your Answer posted. $\endgroup$ – hardmath Jun 20 '16 at 17:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.