If $ A \in B $ and $ B \subseteq C $ then $ A \in C $. vs. If $ A \in B $ and $ B \subseteq C $ then $ A \subseteq C $.

Question

I am trying to decipher the difference between the following two statements:

If $ A \in B $ and $ B \subseteq C $ then $ A \in C $.

vs.

If $ A \in B $ and $ B \subseteq C $ then $ A \subseteq C $.

I think the first statement is the true statement and the 2nd is false because of $ A \in B $ not $A \subseteq B $

Am I on the right track? How would I give a simple counter example to show the statement is false?

Answer 1

That's right. That is pretty much the definition of a subset.

For the second statement consider

$A:=\{1\}$, $B:=\{\{1\},2\}$ and $C:=\{\{1\},2,3\}$.

Then $A\in B$, $B\subseteq C$, but $A\not\subseteq C$, since $1\notin C$.

Answer 2

First statement:

if A is an element in B, and B is a subset of C, then A is in C.

Explanation: this is true because B being a subset of C means that every element in B is also in C.

Second statement:

if A is an element in B and B is a subset of C, then A is a subset of C.

Explanation: this is false because the element A is not necessarily a set. For example, let $C = \{1,2,3\}$. Let $B = \{1,2\}$. Let $A$ be $1$. $1$ is not a subset of $\{1,2,3\}$ -- it is a member.

Hypothetically: $\{1\}$, on the other hand, would be a subset.

Answer 3

Another example. These are true $$ -5 \in \mathbb Z\quad\text{and}\quad \mathbb Z \subseteq \mathbb Q $$ and it follows that $$ -5 \in \mathbb Q $$ but it does not follow that $$ -5 \subseteq \mathbb Q $$