For any sets A and B, prove that:
(i) $(A-B) \cap B=\phi$
(ii) $A \cup(B-A)=A \cup B$
(iii) $(A-B) \cup(A \cap B)=A$
(iv) $(A \cup B)-B=A-B$
(iv) $A-(A \cup B)=A-B$
Two sets are shown with the following Venn Diagram
The yellow region is denoted by 1.
Blue region is denoted by 2.
The common region is denoted by 3.
(i) A - B denotes region 1
B denotes region (2+3)
So their intersection is a : set
$\Rightarrow(A-B)^{\cap} B=\varnothing$
(ii) B - A denotes region 2
A denotes region (1+3)
So their union denotes region (1+2+3) which is the union of A and B
$\Rightarrow \mathrm{A}^{\mathrm{U}}(\mathrm{B}-\mathrm{A})=\mathrm{A}^{\mathrm{U}} \mathrm{B}$
(iii) A - B denotes region 1
$\mathrm{A}^{U} \mathrm{~B}$ denotes region 3
Their union denotes region $(1+3)$ which is set $A \Rightarrow(A-B) \cup_{(A} \cap_{B)}=A$
(iv) $\mathrm{A} \cup \mathrm{B}$ denotes region $(1+2+3)$
$(\mathrm{AUB})-\mathrm{B}$ denotes region $(1+2+3)-(2+3)=1 \mathrm{~A}-\mathrm{B}$ denotes region $1 \Rightarrow\left(\mathrm{A}^{\mathrm{U}} \mathrm{B}\right)-\mathrm{B}=\mathrm{A}-\mathrm{B}$
(v) Wrong question