What is the Set $A_3$ if $A_0=\varnothing$(the empty set).$\forall \space i=1,2,3,...$define the set $A_i=A_{i-1}\cup \{A_{i-1}\}$?

Question

I first used the relation given:

$A_i=A_{i-1}\cup \{A_{i-1}\}$

$\implies A_1=A_0 \cup \{\varnothing\}$

$\therefore \space A_2=A_1 \cup \{A_1\}=\varnothing \cup \{\varnothing\}\cup \{\varnothing \cup\ \{\varnothing\}\}$

$\therefore \space A_3=A_2 \cup \{A_2\}=\varnothing \cup \{\varnothing\}\cup \{\varnothing \cup\ \{\varnothing\}\}\cup \{\varnothing \cup \{\varnothing\}\cup \{\varnothing \cup\ \{\varnothing\}\}\}$

My assumption was the answer is $\{\varnothing\}$. But it was a handwaivy way how I came to this conclusion and comes out to be wrong!

The correct answer to this problem is:

$\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}$

Please give me proper and justifiable answer to this question

Answer 1

You're correct all the way. $$A_3=\emptyset \cup \{\emptyset\}\cup \{\emptyset \cup\ \{\emptyset\}\}\cup \{\emptyset \cup \{\emptyset\}\cup \{\emptyset \cup\ \{\emptyset\}\}\}$$ Simplify the first four $\emptyset$'s. $$A_3=\{\emptyset,\{\emptyset\}\}\cup \{\emptyset \cup \{\emptyset\}\cup \{\emptyset \cup\ \{\emptyset\}\}\}$$ Then the next four, $$A_3=\{\emptyset,\{\emptyset\}\}\cup \{\{\emptyset \cup\ \{\emptyset\}\}\}$$ Finally, just collect the elements: $$A_3=\{\emptyset,\{\emptyset\},\{\emptyset \cup\ \{\emptyset\}\}\}$$

EDIT: Just keep in mind that $S\cup\emptyset=S$, for any set $S$.

Answer 2

Your issue arises from the assertion on line 3:

$\therefore \space A_2=A_1 \cup \{A_1\}=\varnothing \cup \{\varnothing\}\cup \{\varnothing \cup\ \{\varnothing\}\}$

$A_{1} \cup \emptyset $ defines the set which contains the elements of $A_{1}$ and includes the set which contains $\emptyset$. Thus, $A_{1} \cup \emptyset = \{\emptyset,\{\emptyset\}\}.$

Then, just generalize this idea.

The key here is the remember that $\emptyset$ is a set, which means $\{\emptyset\}$ is a set of a set.

Answer 3

$$A_1 = \emptyset \cup \{\emptyset \} = \{\emptyset \} $$

$$A_2 = \{\emptyset\} \cup \{\{\emptyset \}\} = \{\emptyset , \{\emptyset \} \} $$

$$A_3 = \{\emptyset , \{\emptyset \} \} \cup \{\{\emptyset , \{\emptyset \} \} \} = \{\emptyset , \{\emptyset \},\{\emptyset , \{\emptyset \} \}\} $$