# Sets, subsets, and elements

Can anyone help me clear this up.

1. Is {0, 1} ∈ {0, {1}}
2. Is {0, 1} ⊆ {0, {0, 1}}
3. Is b ∈ {a, {a, b}}
4. And finally {a,b} ⊂ {a,a,b}

So I understand that a subset and proper subset (set without line under it) are different, because in a proper subset, A cannot be equal to B if A is a proper subset for B.

I think I'm mostly getting confused with how the set's break down to determine their elements, like for example in problem 1 is the sets elements {0, empty set, {{1}}, {{empty set}}}? So the cardinality would be 2?

• Basic difefrence: $\in$ and $\subseteq$ are different relations. For an element $a$ of a set $A$ we have $a \in A$. For a subset $B$ of a set $A$ we have $B \subseteq A$. – Mauro ALLEGRANZA Oct 4 '17 at 15:49
• $\{ 0,1 \}$ is not and element of $\{ 0, \{ 1 \} \}$ because the first is not "listed" between the elements of the second. The elements of the second are: $0$ and $\{ 1 \}$. – Mauro ALLEGRANZA Oct 4 '17 at 15:51
• only 4 is true and the rest are not – Guy Fsone Oct 4 '17 at 15:51
• The condition for $B$ being a subset of $A$ is that every element of $B$ is also an element of $A$: for every $x$, if $x \in B$, then $x \in A$. – Mauro ALLEGRANZA Oct 4 '17 at 15:52
• The elements of 1. are two: $0$ and $\{ 1 \}$. Thus $\{ 0,1 \}$ is not an element of the set. – Mauro ALLEGRANZA Oct 4 '17 at 15:53