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?
$\endgroup$ 33 Answers
$\begingroup$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 $$ -5 \subseteq \mathbb Q $$
$\endgroup$ $\begingroup$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 $1\notin C$.
$\endgroup$ 3 $\begingroup$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.
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