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 $x \notin B,$ then $x \notin A$.
If $A \subset B$ and $x \notin B,$ then $x \notin A$.
(A) The statement is True.
Given: $A \subset B$ and $x \notin B$.
To prove: $x \notin A$.
Proof by contradiction:
Suppose $x \in A$.
Since $A \subset B$,every element of $A$ must be an element of $B$.
Therefore,$x \in A$ implies $x \in B$.
However,we are given that $x \notin B$.
This is a contradiction.
Therefore,our assumption that $x \in A$ is false.
Hence,$x \notin A$.
Given: $A \subset B$ and $x \notin B$.
To prove: $x \notin A$.
Proof by contradiction:
Suppose $x \in A$.
Since $A \subset B$,every element of $A$ must be an element of $B$.
Therefore,$x \in A$ implies $x \in B$.
However,we are given that $x \notin B$.
This is a contradiction.
Therefore,our assumption that $x \in A$ is false.
Hence,$x \notin A$.
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