In each of the following,determine whether the statement is true or false. If it is true,prove it. If it is false,give a counterexample.
If $A \subset B$ and $B \in C,$ then $A \in C$.
If $A \subset B$ and $B \in C,$ then $A \in C$.
(N/A) The statement is False.
To disprove the statement,we provide a counterexample.
Let $A = \{2\}$,$B = \{0, 2\}$,and $C = \{1, \{0, 2\}, 3\}$.
Here,$A \subset B$ because every element of $A$ (which is $2$) is also an element of $B$.
Also,$B \in C$ because the set $B = \{0, 2\}$ is an element of $C$.
However,$A \notin C$ because the set $A = \{2\}$ is not one of the elements of $C$ (the elements of $C$ are $1$,$\{0, 2\}$,and $3$).
Thus,the statement is false.
To disprove the statement,we provide a counterexample.
Let $A = \{2\}$,$B = \{0, 2\}$,and $C = \{1, \{0, 2\}, 3\}$.
Here,$A \subset B$ because every element of $A$ (which is $2$) is also an element of $B$.
Also,$B \in C$ because the set $B = \{0, 2\}$ is an element of $C$.
However,$A \notin C$ because the set $A = \{2\}$ is not one of the elements of $C$ (the elements of $C$ are $1$,$\{0, 2\}$,and $3$).
Thus,the statement is false.
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