A student made an error while proving the following claim that he thought was true. Claim: For any sets $A$, $B$, $C$, we have $A \backslash ( B \cap C ) \subseteq ( A \backslash B ) \cap ( A \backslash C )$. The student's proof: If I show that every element of the set $A \backslash ( B \cap C )$ is in the set $( A \backslash B ) \cap ( A \backslash C )$, the proof is complete. Now, let $x \in A \backslash ( B \cap C )$. (I) From this, $x \in A$ and $x \notin ( B \cap C )$. (II) From this, $x \in A$ and $( x \notin B$ and $x \notin C )$. (III) From this, $( x \in A$ and $x \notin B )$ and $( x \in A$ and $x \notin C )$. (IV) From this, $x \in A \backslash B$ and $x \in A \backslash C$. (V) From this, $x \in [ ( A \backslash B ) \cap ( A \backslash C ) ]$. In which of the numbered steps did this student make an error? A) I B) II C) III D) IV E) V
A student made an error while proving the following claim that he thought was true.
Claim: For any sets $A$, $B$, $C$, we have $A \backslash ( B \cap C ) \subseteq ( A \backslash B ) \cap ( A \backslash C )$.
The student's proof:
If I show that every element of the set $A \backslash ( B \cap C )$ is in the set $( A \backslash B ) \cap ( A \backslash C )$, the proof is complete.
Now, let $x \in A \backslash ( B \cap C )$.\\
(I) From this, $x \in A$ and $x \notin ( B \cap C )$.\\
(II) From this, $x \in A$ and $( x \notin B$ and $x \notin C )$.\\
(III) From this, $( x \in A$ and $x \notin B )$ and $( x \in A$ and $x \notin C )$.\\
(IV) From this, $x \in A \backslash B$ and $x \in A \backslash C$.\\
(V) From this, $x \in [ ( A \backslash B ) \cap ( A \backslash C ) ]$.
In which of the numbered steps did this student make an error?
A) I\\
B) II\\
C) III\\
D) IV\\
E) V