Q. Let A and B are two sets in a universal set U. Then which of these is/are correct ?
Sets
Solution:
$\left(a\right) x\in A -B \Leftrightarrow x \in A$ and $x \notin B$
$\Leftrightarrow x\in A$ and $x\in B' \Leftrightarrow x\in A \cap B'$
$\therefore A - B = A \cap B' \quad...\left(i\right)$
$x \in A$ and $x \in B'$
$\Leftrightarrow x \notin A'$ and $x \in B' \Leftrightarrow x \in B'$ and $x \notin A'$
$\Leftrightarrow x \in B' -A'$
$\therefore A - B = B' - A'\quad ...\left(ii\right)$
Clearly $\left(a\right)$ is not correct. Also from $\left(i\right) \left(c\right)$ is not correct.
Next let $x\in A - \left(A - B\right)$
$\Leftrightarrow x\in A$ and $x \notin A -B$
$\Leftrightarrow x\in A$ and [$x \notin A$ or $x \in B$]
$A - (A - B) = A \cap B$
$\Leftrightarrow$ [$x\in A$ and $x \notin A$] or [$x\in A$ and $x \in B$]
$\therefore A - \left(A - B\right) = \phi \cup \left(A \cap B\right) = A \cap B$
$\therefore \left(b\right)$ is also incorrect
The result $\left(d\right)$ is correct as can be seen in the following Venn diagram
$A \cup B = \left(A - B\right) \cup \left(A \cap B\right) \cup \left(B - A\right)$
