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 $x \in A$ and $A \in B,$ then $x \in B$.
If $x \in A$ and $A \in B,$ then $x \in B$.
(N/A) The statement is False.
To disprove the statement,we provide a counterexample.
Let $A = \{1, 2\}$ and $B = \{3, 4, \{1, 2\}\}$.
Here,$1 \in A$ and $A \in B$ (since $\{1, 2\}$ is an element of $B$).
However,$1 \notin B$ because $1$ is not one of the elements $\{3, 4, \{1, 2\}\}$.
Thus,the statement is false.
To disprove the statement,we provide a counterexample.
Let $A = \{1, 2\}$ and $B = \{3, 4, \{1, 2\}\}$.
Here,$1 \in A$ and $A \in B$ (since $\{1, 2\}$ is an element of $B$).
However,$1 \notin B$ because $1$ is not one of the elements $\{3, 4, \{1, 2\}\}$.
Thus,the statement is false.
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