set of $Ф$ as $\{ \}$ proof of uniqueness

Question

The following set: $S = \{Ф, \{Ф\}, \{Ф, \{Ф\}\}\}$ can be represented by the following string (finite sequence) of “$\{$” and “$\}$”: $s = \{ \{ \} \{ \{ \} \} \{ \{ \} \{ \{ \} \} \} \}$ Note that the empty set $Ф$ was written as $\{ \}$ and all commas were omitted. $1)$ Show that if a string s of “$\{$“ and “$\}$” represents a set $S$, then $S$ is unique, i.e. two different sets are represented by two different strings.

Answer 1

Hint: From a (suitable) string of $\{$s and $\}$s, you can recover the expression a finite set written in list notation by first replacing all instances of "$\}\{$" by "$\},\{$", and then replacing all instances of "$\{\}$" by "$\varnothing$". Now use the fact that two sets are equal if and only if they contain the same elements.