Translation from Dutch

Question

Google Translator cannot help me with this translation. May you translate it?

Answer 1

6. What is the connection between the following three properties of a subset $A$ of $\mathbb{R}$?

(a) $A$ is open. (b), (c) are pure maths.

If you say one property implies another, you have to prove that implication. If you say that a property does not imply another, you have to give an explicit example to show that.

7. Consider the set $\mathbb{I}$ of irrational numbers [personal note: $\mathbb{P}$ is actually more common]. Is $\mathbb{I}$ an open subset of $\mathbb{R}$? Is $\mathbb{I}$ a closed subset of $\mathbb{R}$? Give an argument for that. [so: give proofs, not just answers]

Answer 2

6. What is the connection between the following properties of subset $A\subseteq\mathbb R$?

   • (a) $A$ is open.
   • (b) as in original
   • (c) as in original

If you assert that one of these properties implies another of them then a proof of that is needed. If you assert that one of these properties does not imply another of them then a concrete illustrating example of this is requested.

7. Let $\mathbb I\subseteq\mathbb R$ denote the set of irrational numbers. Is $\mathbb I$ an open set of $\mathbb R$? Is it a closed set? Give arguments for that.

Answer 3

Automatic translations tend to deliver surprisingly good results.

Find below a comparison of the translations provided by Google Translate and by DeepL Translator:

Input text:

6. Wat is het verband tussen onderstaande eigenschappen voor een deel XXX? (a) ... (b) ... (c) ... Als je zegt dat de ende eigenschap de andere impliceert moet je dat bewijzen. Als je zegt dat de ene eigenschap de andere niet impliceert moet je dat illustreren met een concreet voorbeeld.
7. Beschow de verzameling X van den irrationale getallen als deel van R. Is I een open deel van R? Is I een gesloten deel van R? Argumenteer.

Output of DeepL Translator:

6. What is the relationship between the properties below for part XXX? (a) ... (b) ... (c) ... If you say that the property implies the other, you must prove it. If you say that one property does not imply the other, you should illustrate this with a concrete example.
7. Consider the set X of irrational numbers as part of R. Is I an open part of R? Is I a closed part of R? Argument.

Output of Google Translate:

6. What is the relationship between the characteristics below for part XXX? (a) ... (b) ... (c) ... If you say that the one property implies the other you have to prove it. If you say that one property does not imply the other you must illustrate it with a concrete example.
7. Do the collection X of the irrational numbers as part of R. Is I an open part of R? Is I a closed part of R? Argument.