I'm having difficulty understanding the difference between $\displaystyle \in$ and $\displaystyle \subset$
A question asks "true or false? $\displaystyle \{1,2 \} \in \{1,2, \{ \{1,2 \} \} \}$ I thought this was true but it is false.
I'm having difficulty understanding the difference between $\displaystyle \in$ and $\displaystyle \subset$
A question asks "true or false? $\displaystyle \{1,2 \} \in \{1,2, \{ \{1,2 \} \} \}$ I thought this was true but it is false.
1, 2, 1, 2 is not a set, so I don't know what you're referring to. Do you mean {1, 2} and {1, 2, 1, 2}? Those are sets, and they are the same set because they contain the same elements. They would also be subsets of each other.
{1} is a subset of {1, 2}. 1 is a member of {1, 2}.
For the set {1, 2} to be inside another set, you'd have to see {1, 2} or its equivalent inside that set.
{ {1, 2} } contains {1, 2}.
{1, 2} does not contain {1, 2}. It is {1, 2}, however.
{1, 2, {{1, 2}}} does not contain {1, 2}. It has three members. We just remove the outer braces to list them: 1, 2, and {{1, 2}}. Of these members, {{1, 2}} does indeed contain {1, 2}, but the enclosing set does not.
Think of membership as "contains" and subset as "includes" and this might help.