Let $U$ be the set of all boys and girls in a school, $G$ be the set of all girls in the school, $B$ be the set of all boys in the school and $S$ be the set of all students in the school who take swimming. Some but not all, students in the school take swimming. Draw a Venn diagram showing one of the possible interrelationship among sets $U, G, B$ and $S$.
For all sets $A, B$ and $C$, show that $(A-B) \cap(A-C)=A-(B \cup C)$.
$$\begin{aligned} & \text { Let } \quad x \in(A-B) \cap(A-C) \\ & \Rightarrow \quad x \in(A-B) \text { and } x \in(A-C) \\ & \Rightarrow \quad(x \in A \text { and } x \notin B) \text { and }(x \in A \text { and } x \notin C) \\ & \Rightarrow \quad x \in A \text { and }(x \notin B \text { and } x \notin C) \\ & \Rightarrow \quad x \in A \text { and } x \notin(B \cup C) \\ & \Rightarrow \quad x \in A-(B \cup C) \\ & \Rightarrow \quad(A-B) \cap(A-C) \subset A-(B \cup C) \quad \text{.... (i)}\\ & \text { Now, let } \quad y \in A-(B \cup C) \\ & \Rightarrow \quad y \in A \text { and } y \notin(B \cup C) \\ & \Rightarrow \quad y \in A \text { and }(y \notin B \text { and } y \notin C) \\ & \Rightarrow \quad(y \in A \text { and } y \notin B) \text { and }(y \in A \text { and } y \notin C) \\ & \Rightarrow \quad y \in(A-B) \text { and } y \in(A-C) \\ & \Rightarrow \quad y \in(A-B) \cap(A-C) \\ & \Rightarrow \quad A-(B \cup C) \subset(A-B) \cap(A-C) \quad \text{.... (ii)}\\ & \text { From Eqs. (i) and (ii), } \\ & A-(B \cup C)=(A-B) \cap(A-C) \end{aligned}$$
For all sets $A$ and $B,(A-B) \cup(A \cap B)=A$.
$\begin{aligned} \mathrm{LHS} & =(A-B) \cup(A \cap B) \\ & =[(A-B) \cup A] \cap[(A-B) \cup B] \\ & =A \cap(A \cup B)=A=\mathrm{RHS}\end{aligned}$
Hence, given statement is true.
For all sets $A, B$ and $C, A-(B-C)=(A-B)-C$.
See the Venn diagrams given below, where shaded portions are representing $A-(B-C)$ and $(A-B)-C$ respectively.
Clearly, $$ A-(B-C) \neq(A-B)-C . $$ Hence, given statement is false.
For all sets $A, B$ and $C$, if $A \subset B$, then $A \cap C \subset B \cap C$.
$$\begin{array}{ll} \text { Let } & x \in A \cap C \\ \Rightarrow & x \in A \text { and } x \in C \\ \Rightarrow & x \in B \text { and } x \in C \quad [\because A \subset B]\\ \Rightarrow & x \in(B \cap C) \Rightarrow(A \cap C) \subset(B \cap C) \end{array}$$
Hence, given statement is true.