# Is $\{\varnothing \}$ an empty set?

Is $\{\varnothing \}$ an empty set ? this suppose to be 7th grade math ,i went through the empty set lesson in the textbook , basically i know that

{} or $\varnothing$ is an empty set

but what about $\{\varnothing\}$ which is a question in the textbook , i was thinking

what if $\varnothing$ is a symbol in $\{\varnothing\}$

but again im not sure . i'll appreciate any help . thanks for advance.

• Think of it like this: a box that contains an empty box isn't empty, is it? – MPW Sep 2 '14 at 21:31
• that is a very good hint ! wow! thank you so much – Bswan Sep 2 '14 at 21:33

HINT: Recall that whatever $x$ is, $x\in\{x\}$.

• i edited the question , so it wont change the answer that its not an empty set right ? – Bswan Sep 2 '14 at 21:01
• My hint still applies. Now pick $x=\varnothing$ instead of $x=\phi$. – Asaf Karagila Sep 2 '14 at 21:01
• just wanted to make sure :P . thank you so much again , you have no idea how it was helpful !i understand it now. – Bswan Sep 2 '14 at 21:04
• No problem. I'm glad I could help! – Asaf Karagila Sep 2 '14 at 21:04

Note that there is only one empty set. It is the only set which has no elements. Thus, it is not meaningful (and a sign that you haven't fully grasped the concept yet) if you talk about an empty set in your question.

$\emptyset$ and $\{\}$ are only two different symbolic representations for the same thing - like e.g. "$4$" and the Roman numeral "IV" are two different representations for the same number.

• i think i got it , i only had adoubt about it. thank you :) – Bswan Sep 2 '14 at 20:53

one denotes the empty set by $\emptyset$ or $\{ \}$.

The set $\{\phi\}$ is not empty as it contains an element, namely $\phi$.

Note the difference between the LaTeX symbols: $\phi$ and $\emptyset$.

• the symbol that im asking for is phi , which would be the empty set notation if its alone :) thank you . – Bswan Sep 2 '14 at 20:50
• @Bswan usually one uses $\emptyset$ or $\varnothing$ (the latter being my preferred choice) rather than the letter $\phi$ which is not standard for the empty set, they just look similar on paper. – Adam Hughes Sep 2 '14 at 20:54