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Q. Determine which of the following are true?
I. If $x \in A$ and $A \in B$, then $x \in B$.
II. If $A \subset B$ and $B \in C$, then $A \in C$.
III. If $A \subset B$ and $B \subset C$, then $A \subset C$.
IV. If $A \not \subset B$ and $B \not \subset C$, then $A \not \subset C$.
V. If $x \in A$ and $A \not \subset B$, then $x \in B$.
VI. If $A \subset B$ and $x \notin B$, then $x \notin A$.

Sets

Solution:

I. If $x \in A$ and $A \in B \Rightarrow x \notin B$
As $A=\{1,2\}, B=\{\{1,2\}, 3\}$
Then, $ 1 \in A$ but $1 \notin B$
II. If $A \subset B$ and $B \in C \Rightarrow A \notin C$
As $A=\{1\}, B=\{1,2\}$ and $C=\{\{1,2\}, 3\}$
III. If $A \subset B$ and $B \subset C$
$\Rightarrow$ Every element of $A$ is in $B$ and every element of $B$ is in $C$
$\Rightarrow$ Every element of $A$ is also in $C$.
$\Rightarrow A \subset C$
IV. If $A \not \subset B$ and $B \not \subset C$, then $A$ may be subset of $C$.
As $A=\{1,2\}, B=\{3\}, C=\{1,2,4,5\}$
V. If $x \in A$ and $A \not \subset B \Rightarrow x \notin B$
As $A=\{1,2\}, B=\{3,4,5\}$
Here, $1 \in A$ and $A \not \subset B$ but $1 \notin B$
VI. If $A \subset B$ and $x \notin B$, then $x \notin A$
As $A \subset B$
$\Rightarrow$ Every element of $A$ is in $B$.
i.e., there does not exist any element of $A$ which does not belong to $B$.