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Which of the following statements are true?

(a) Every connected subset of $\mathbb{R^n}$ is convex

(b) Every convex subset of $\mathbb{R^n}$ is compact

(c) Every convex subset of $\mathbb{R^n}$ is connected

(d) The set $\left\{ (x,y) \in \mathbb{R}^2 : x \geq 0, y \geq 0 \right\} \cup \left\{ (x,y) \in \mathbb{R}^2 : x \leq 0, y \leq 0 \right\}$ is convex


It is given that option $c$ is correct. I agree if the set is convex then clearly it is connected too. And connected need not be compact and convex set also need not be compact so option (a) and (b) are false. But, what about option $d$? I think it should also be correct, as the union is the entire $\mathbb{R}^2$ plane, it must be convex. Am I correct?

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  • $\begingroup$ now completed, thankyou for reminding... $\endgroup$ Commented Nov 30, 2023 at 16:20
  • $\begingroup$ If $(x,y)\in\mathbb{R}^2$ such that either $x<0$, $y>0$ is true, or $x>0$, $y<0$ is true, then $(x,y)$ is not the the union, so clearly it cannot be all of $\mathbb{R}^2$. Also $(0,2),(-2,0)$ are in the union, but the convex combination $(-1,1)=\frac{1}{2}(0,2)+\left(1-\frac{1}{2}\right)(-2,0)$ is clearly not, so it's not convex $\endgroup$
    – Lorago
    Commented Nov 30, 2023 at 16:41
  • $\begingroup$ Actually this is from a competition exam... $\endgroup$ Commented Dec 2, 2023 at 2:16
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    $\begingroup$ its an competitive exam for teachers, to be eligible for the post of assistant professor.. $\endgroup$ Commented Dec 3, 2023 at 6:44

1 Answer 1

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The set described in d) is not $\mathbb{R}^2$, it is the union of 1st and 3rd quadrants of the cartesian plane. It is connected because $0$ belongs to the set but it is clearly not convex

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