Question

Consider the following statements for non empty sets A, B and C
1 $A-(B-C)=(A-B)\cup C$
2 $A-(B\cup C)=(A-B)-C$
which of the statements given above is/are correct?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

The correct option is B 2 only

1. $A-(B-C)=A-(B\cap C^{\prime })$
$=A\cap (B\cap C^{\prime }{)}^{\prime }$
$=A\cap (B^{\prime }\cup (C^{\prime }\left)\right)$
$=A\cap (B^{\prime }\cup C)$
Thus, $A-(B-C)\ne (A-B)\cup C$

2. A- $(B\cup C)=A\cap (B\cap C{)}^{\prime }$
$=A\cap (B^{\prime }\cap C{)}^{\prime }$
$(A-B)-C=(A\cap B^{\prime })-C$
$=A\cap B^{\prime }\cap C^{\prime }$
$\Rightarrow A-(B\cup C)=(A-B)-C$
Associative property.