# Describing set with union/intersection summation

I am trying to describe $$\bigcup_{i=1}^\infty A_i$$ and $$\bigcap_{i=1}^\infty A_i$$ for the following two sets, but I am having a hard time conceptualizing it. $$(a)\ A_i = \{-i, -i+1,\ ...\ ,-1,0,1,\ ...\ ,i-1,i\}$$$$(b)\ A_i = \{-i, i\}$$ I think that for the set $$(a)$$, $$\bigcup_{i=1}^\infty A_i$$ represents any arbitrary set of integer, and $$\bigcap_{i=1}^\infty A_i$$ represents the set of all integers. However I'm not sure that this is correct, and I am having a hard time describing the union and intersection summations for $$(b)$$.

EDIT: formatting

• What do you mean by represents any arbitrary set of integer? $\cup A_i$ is just one set. – mathcounterexamples.net Jun 30 '19 at 19:58
• @mathcounterexamples.net I was saying that because I was treating $\cup$ like the $\lor$ operator. I'm still struggling to conceptualize this. – Robert Schwartz Jun 30 '19 at 20:06

Let’s denote in the two cases $$A = \bigcup_{i=1}^\infty A_i$$ and $$B=\bigcap_{i=1}^\infty A_i$$
(a) $$\bigcup_{i=1}^\infty A_i = \mathbb Z$$. Indeed any integer number $$n$$ belongs to $$A_n$$ and therefore to $$A$$, proving that $$\mathbb Z \subseteq \bigcup_{i=1}^\infty A_i$$. Conversely, every set $$A_i$$ is a set of integers, hence their union two. Therefore $$\bigcup_{i=1}^\infty A_i \subseteq \mathbb Z$$.
$$\bigcap_{i=1}^\infty A_i = \{0\}$$.
As $$0$$ belongs to all $$A_i$$ it belongs to their intersection. And none other integer belongs to this intersection. Which proves the equality.
(b) You have $$\bigcup_{i=1}^\infty A_i = \mathbb Z \setminus \{0\}$$ and $$\bigcap_{i=1}^\infty A_i = \emptyset$$. I left the proof to you.