In each of the following,determine whether the statement is true or false. If it is true,prove it. If it is false,give an example.
If $A \subset B$ and $B \subset C$,then $A \subset C$.
If $A \subset B$ and $B \subset C$,then $A \subset C$.
(A) The statement is True.
Proof:
Let $A \subset B$ and $B \subset C$.
Let $x$ be an arbitrary element such that $x \in A$.
Since $A \subset B$,by the definition of a subset,$x \in B$.
Since $B \subset C$,by the definition of a subset,$x \in C$.
Since $x \in A$ implies $x \in C$,it follows that $A \subset C$.
Proof:
Let $A \subset B$ and $B \subset C$.
Let $x$ be an arbitrary element such that $x \in A$.
Since $A \subset B$,by the definition of a subset,$x \in B$.
Since $B \subset C$,by the definition of a subset,$x \in C$.
Since $x \in A$ implies $x \in C$,it follows that $A \subset C$.
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