Question

Let A and B are two sets in a universal set U. Then which of these is/are correct ?

• A. $A-B=A^{\prime }-B^{\prime }$
• B. $A-(A-B)=A\cup B$
• C. $A-B=A^{\prime }\cap B^{\prime }$
• D. $A\cup B=(A-B)\cup (B-A)\cup (A\cap B)$

Answer 1

Answer: (D)

$\left(a\right)x\in A-B\iff x\in A$ and $x\notin B$
$\iff x\in A$ and $x\in B^{\prime }\iff x\in A\cap B^{\prime }$
$\therefore A-B=A\cap B^{\prime }...\left(i\right)$
$x\in A$ and $x\in B^{\prime }$
$\iff x\notin A^{\prime }$ and $x\in B^{\prime }\iff x\in B^{\prime }$ and $x\notin A^{\prime }$
$\iff x\in B^{\prime }-A^{\prime }$
$\therefore A-B=B^{\prime }-A^{\prime }...\left(ii\right)$
Clearly $\left(a\right)$ is not correct. Also from $\left(i\right)\left(c\right)$ is not correct.
Next let $x\in A-\left(A-B\right)$
$\iff x\in A$ and $x\notin A-B$
$\iff x\in A$ and [$x\notin A$ or $x\in B$]
$A-(A-B)=A\cap B$
$\iff$ [$x\in A$ and $x\notin A$] or [$x\in A$ and $x\in B$]
$\therefore A-\left(A-B\right)=\varphi \cup \left(A\cap B\right)=A\cap B$
$\therefore \left(b\right)$ is also incorrect
The result $\left(d\right)$ is correct as can be seen in the following Venn diagram
$A\cup B=\left(A-B\right)\cup \left(A\cap B\right)\cup \left(B-A\right)$