Prove or give a counterexample: If $A$ is a subset of $B$ and $B$ belongs to $C$, then $A$ belongs to $C$.

Question

Prove or give a counterexample: If $A$ is a subset of $B$ and $B$ belongs to $C$, then $A$ belongs to $C$.

I think this is false because of the counterexample

$$ \begin{align} A &= \{1,2\}\\ B &= \{1,2,3\}\\ C &= \{\{1,2,3\}\} \end{align} $$

but I am not sure if I am right.

Answer 1

Nice counterexample, you are correct. Let us check each condition:

$A$ is a subset of $B$

That's true: you have $A = \{1,2\}$ and $B = \{1,2,3\}$. The elements of $A$ are $1$ and $2$, and both of them are also elements of $B$.

$B$ belongs to $C$

That's true also. You took $C = \{\{1,2,3\}\}$. $C$ has one element, and that's $B$. $C = \{B\}$.

then $A$ belongs to $C$

This is false -- $C$ only has one element, $B$. But $B$ isn't $A$, becuase $B$ has $3$ elements and $A$ only has $2$ elements. Specifically, $B$ has $3$ and $A$ doesn't.

So your counterexample is correct: $A$ is a subset of $B$, and $B$ belongs to $C$, but that doesn't necessarily imply that $A$ belongs to $C$.