Union and Intersection of Family of Sets

Question

I am lost on understanding this concept. Let $\lbrace A_n\mid n \in \mathbb{N} \rbrace$ be an infinite family of sets where $A_n = \lbrace 0, 1, 2, \ldots, n\rbrace$. Consider the universal set here to be $\mathbb{N}$. How would I find the union up to $10$ and the intersection of up to $10$. I cannot find anything that explains the concept very well.

Answer 1

The key to this problem is to use
for all $n,m \in \mathbb{N}, (n \le m \to A_n \subseteq A_m)$, and
for all $A,B$ ($A \subseteq B$ implies $A \cap B = A$ and $A \cup B = B)$.

Answer 2

Let's try to think about this pictographically:

Draw the set $A_2 = \{0,1,2\}$ as a diagram with those elements floating around in it.

In set theory two sets are considered the same if they have the same elements. Similarly a set is a subset of another if every element of the first is an element of the second.

Therefore it is clear that $A_2 \subset A_3$ and in fact you should try to formulate a proof of $A_n \subset A_{n+k}$ where $k \geq 0$.

Now draw $A_3$. From the above discussion it should be clear that it contains $A_2$ and differs from it only in containing the element $3$.

$A \cap B$ is a symbolised way of saying the set of elements in both $A$ and $B$. Therefore, what's $A_2 \cap A_3$?

$A \cup B$ is asking you what do you get by combining these sets.

Again from our discussion it should be clear that $A_2 \cap A_3 = A_2$ and $A_2 \cup A_3 = A_3$ (since we don't count elements twice in set theory).

Try to generalize this for arbitrary finite unions and intersections to get your answer.

Hope this helps,