Question

Let $A,B$ are two non-empty sets and $U$ be the universal set. Then which of the following statements is/are true?
$1.n(A\cup B{)}^{\prime }=n(A^{\prime }\cap B^{\prime })$
$2.$ If $A\cap B=\varphi ,$ then $A^{\prime }\cup B^{\prime }=U$
$3.$ If $A\cup B=U,$ then $A^{\prime }\cap B^{\prime }=\varphi$
$4.$ If $A\subset B,$ then $A^{\prime }\cup B^{\prime }=(A\cap B{)}^{\prime }$
(where $\varphi$ denote the null set)

• A. $1,2$ and $3$ only
• B. $1$ and $3$ only
• C. $1,2$ and $4$ only
• D. All statements are true

Answer 1

The correct option is D All statements are true

$1.n(A\cup B{)}^{\prime }=n(A^{\prime }\cap B^{\prime })$
This is true by De Morgan's law.

$2.$ If $A\cap B=\varphi ,$ then $A^{\prime }\cup B^{\prime }=U$
$A^{\prime }\cup B^{\prime }=(A\cap B{)}^{\prime }=U-(A\cap B)=U$
This statement is correct.

$3.$ If $A\cup B=U,$ then $A^{\prime }\cap B^{\prime }=\varphi$
$A^{\prime }\cap B^{\prime }=(A\cup B{)}^{\prime }=U-(A\cup B)=U-U=\varphi$
This statement is correct.

$4.$ If $A\subset B,$ then $A^{\prime }\cup B^{\prime }=(A\cap B{)}^{\prime }$
$A\subset B\Rightarrow A\cap B=A$
$\Rightarrow (A\cap B{)}^{\prime }=A^{\prime }$

$A\subset B\Rightarrow B^{\prime }\subset A^{\prime }$
$\Rightarrow A^{\prime }\cup B^{\prime }=A^{\prime }$
$\therefore A^{\prime }\cup B^{\prime }=(A\cap B{)}^{\prime }$
This statement is true.

Hence, all the four statements are true.