Which one of the following statements is correct regarding the elements and subsets of the set

Question

Question

Which one of the following statements is correct regarding the elements and subsets of the set $\left \{1, 2, \left \{1, 2, 3 \right \}\right \}$

1. $\left \{1, 2,\right \} \epsilon \left \{1, 2, \left \{1, 2, 3 \right \}\right \}$

2. $\left \{1, 2\right \}\subseteq \left \{1, 2, \left \{1, 2, 3 \right \}\right \}$

3. $\left \{1, 2,3 \right \}\subseteq \left \{1, 2, \left \{1, 2, 3 \right \}\right \}$

4. $ 3 \epsilon \left \{1, 2, \left \{1, 2, 3 \right \}\right \}$

I think that $2$ should be the answer because $\left \{1, 2 \right \}$ is subset of $\left \{1, 2, \left \{1, 2, 3 \right \}\right \}$ and $1,2$ are elements of $\left \{1, 2, \left \{1, 2, 3 \right \}\right \}$

i.e $1,2 \epsilon \left \{1, 2, \left \{1, 2, 3 \right \}\right \}$

Am i correct?

Answer 1

If you are using $\subseteq$ and $\in$ very formally then only 2. is correct for the reason you listed. However, $\{1,2,3\}\in\{1,2\{1,2,3\}\}$, but depending on how you have defined $\subseteq$ it may or may not be a subset. This is really a question about the definition of $\subseteq$ and $\in$, which varies from author to author somewhat.

Answer 2

I agree with you. 1) is meaningless,and 4) is incorrect, as far as I can see. As for 3), $\{1,2,3\}\in\{1,2,\{1,2,3\}\}$ would be the correct statement.