# Arbitrary Union of Sets Question

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1. Mar 12, 2016

### TyroneTheDino

1. The problem statement, all variables and given/known data

For each $n \in \mathbb{N}$, let $A_{n}=\left\{n\right\}$. What are $\bigcup_{n\in\mathbb{N}}A_{n}$ and $\bigcap_{n\in\mathbb{N}}A_{n}$.

2. Relevant equations

3. The attempt at a solution
I know that this involves natural numbers some how, I am just confused on a notation thing.
Is $\bigcup_{n\in\mathbb{N}}A_{n}= \left \{ 1,2,3,... \right \}$ or$\bigcup_{n\in\mathbb{N}}A_{n}= \left \{ \left\{1\right\},\left\{2\right\},... \right \}$.

To me it makes more sense that this is set of each set of every natural numbers, but somehow I think this is wrong and can't wrap my head around it.

Last edited: Mar 12, 2016
2. Mar 12, 2016

### geoffrey159

The problem statement is incomplete, it doesn't describe $A_n$.

In terms of notation,
• $x \in \cup_{n\in\mathbb{N}} A_n$ if and only if there exists $n\in\mathbb{N}$ such that $x\in A_n$ ($x$ belongs to at least one of the $A_n$).
• $x \in \cap_{n\in\mathbb{N}} A_n$ if and only if $x\in A_n$ for all $n\in\mathbb{N}$ ($x$ belongs to all $A_n$).

3. Mar 12, 2016

### TyroneTheDino

I updated it to define An.

4. Mar 12, 2016

### Staff: Mentor

$\bigcup_{n\in\mathbb{N}}A_{n}$ means $A_1 \cup A_2 \cup \dots \cup A_n$ and similar for the intersection.

5. Mar 12, 2016

### geoffrey159

Ok so you infered that the union was equal to $\mathbb{N}$. How do you prove that two sets are equal ?