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 $A \not\subset B$ and $B \not\subset C$,then $A \not\subset C$.
If $A \not\subset B$ and $B \not\subset C$,then $A \not\subset C$.
(N/A) The statement is False.
To disprove the statement,we need to find a counterexample where $A \not\subset B$ and $B \not\subset C$ are true,but $A \subset C$ is also true.
Let $A = \{1, 2\}$,$B = \{3, 4\}$,and $C = \{1, 2, 5\}$.
Here,$A \not\subset B$ (since $1 \in A$ but $1 \notin B$) and $B \not\subset C$ (since $3 \in B$ but $3 \notin C$).
However,$A \subset C$ because every element of $A$ is also an element of $C$.
Thus,the statement is false.
To disprove the statement,we need to find a counterexample where $A \not\subset B$ and $B \not\subset C$ are true,but $A \subset C$ is also true.
Let $A = \{1, 2\}$,$B = \{3, 4\}$,and $C = \{1, 2, 5\}$.
Here,$A \not\subset B$ (since $1 \in A$ but $1 \notin B$) and $B \not\subset C$ (since $3 \in B$ but $3 \notin C$).
However,$A \subset C$ because every element of $A$ is also an element of $C$.
Thus,the statement is false.
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