Let $A, B$ and $C$ be three sets. If $A ∈ B$ and $B ⊂ C$, is it true that $A ⊂ C?$If not, give an example.
This question is from my textbook. And the answer is:
No. Let $A = \{1\}, B = \{\{1\}, 2\}$ and $C = \{\{1\}, 2, 3\}.$ Here $A ∈ B$ as $A = {1}$ and $B ⊂ C.$ But $A ⊄ C$ as $1 ∈ A$ and $1 ∉ C.$ Note that an element of a set can never be a subset of itself.
I am having trouble in in understanding "But $A ⊄ C$ as $1 ∈ A$ and $1 ∉ C."$ How could $1 ∉ C?$ Clearly $C$ contains $B$ and $A$ is an element of $B$ and $A$ have $1$ so $C$ must have $1$.
I would be very grateful If you answer this.
Best Answer
This is becuase there is a distinction between $1$ and $\{1\}$. The former is the number one. The latter is a set containing the number one. If some set $C$ contains another set, let's call it set $E$, we do not look at the members of $E$ when we consider the members of $C$. We would say the set $E$ is a member of $C$, but this does not necessarily mean that anything contained by $E$ is also in $C$.