Arbitrary Union of Sets Question

[TyroneTheDino]

Homework Statement

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

Homework Equations

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.

 

[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 $A_n$).

 

[TyroneTheDino]

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 $A_n$).

I updated it to define An.

 

[Mark44]

Homework Statement

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

Homework Equations

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$\bigcup_{n\in\mathbb{N}}A_{n}= \left \{ \left\{1\right\},\left\{2\right\},... \right \}$.

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

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.

 

[geoffrey159]

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