Find the different set.

Question

Find the different set:

(i) $\ \ (\emptyset \cup \{\emptyset\})\cup \{0\}$

(ii) $\ \{\emptyset , \{\emptyset\} , 0\}$

(iii) $\{\emptyset, 0 \}$

I think it is (iii), as (i) & (ii) are same, but the correct answer is (ii). Why?

Answer 1

Since $\emptyset$ has no elements, then $\emptyset \cup \{\emptyset\} = \{\emptyset\}$. Therefore, (i) and (iii) are equal, because

\begin{align} (\emptyset \cup \{\emptyset\}) \cup \{0\} = \{\emptyset\} \cup \{0\} =\{\emptyset, 0\}. \end{align}

Answer 2

(i) and (iii) are the same. Let's take a look at (i):

Ignoring the ")", the first set is a Union of emptyset and a set that contains the emptyset, resulting in a set that contains the emptyset. Now we have a Union with the second set containing "$0$", resulting in (iii).

What is important here: $\emptyset \cup \{ \emptyset \}$ is not the same as $\emptyset , \{ \emptyset \}$. Realize that the first expression is giving you one element, second is giving you two.