Understanding Sets in Real Analysis

[phillyolly]

Homework Statement

The Attempt at a Solution

The solution at the end of the book says that the answer for a) is A5. Why is it so?

Please also explain me the meaning for the question b).

 

[Raskolnikov]

Part (a):
$$A_1 = \{2k : k \in\textbf{N}\}$$ and $$A_2 = \{3k : k \in\textbf{N}\}.$$
So $$A_1 \cap A_2 = \{x : x = 2k_1 \ \wedge \ x = 3k_2 , k_1 \in \textbf{N} , k_2 \in\textbf{N}\}.$$ In plainer words, this set contains all natural numbers that are divisible both by 2 and by 3, i.e., they are divisible by 6. Do you understand the rest?

Part (b):
I don't want to give it entirely away, so I'll suggest looking at some examples.
e.g. Let n = 5. Then $$\bigcup\{A_n : n \in\textbf{N}\}$$ is the set of natural numbers divisible by 2 or 3 or 4 or 5 or 6. And $$\bigcap\{A_n : n \in\textbf{N}\}$$ is the set of natural numbers divisible by 2 and 3 and 4 and 5 and 6.

 

[phillyolly]

Thank you,
I have understood a).
May I ask why did you pick n=5 in b)? I understand it is an example. But I don't understand its connection to the problem.
I am dumb, I know. Sorry.

 

[Raskolnikov]

Nah, I just chose n = 5 as a random example. I could've used n = 3, 4, 17, 9018, ... doesn't matter. When you're having trouble understanding the generalized form of a problem/proof, it often helps to look at individual cases. There's no real pretty/concise way of expressing either of the sets in (b) that I can think of, but it's important you can at least describe them to yourself in words.

i.e., "For any natural number n, $$\bigcup\{A_n : n \in\textbf{N}\}$$ is just the set of natural numbers that are ... and $$\bigcap\{A_n : n \in\textbf{N}\}$$ is just the set of natural numbers that are ... "