For any two sets A and B, prove that (A-B) (B )=A - Vidyadip

Question

For any two sets A and B, prove that
$\text{(A}-\text{B)}\cup(\text{B}\cap\text{A})=\text{A}$

✓

Answer

Let $\text{x}\in\text{A.}$
Then either $\text{x} \in \text{(A}- \text{B) or x} \in \text{(A}\cap \text{B})$
$\Rightarrow \text{x}\in\text{(A – B)}\cup \text{(A} \cap\text{B})$
$\therefore\text{A} \subset \text{(A – B)}\cup \text{(A}\cap\text{B}).....\text{(i)}$
Conversely,
Let $\text{x}\in\text{(A} - \text{B)} \cup \text{(A}\cap\text{B)}$
$\Rightarrow\text{x} \in \text{(A – B) or x (A} \cap \text{B})$
$\Rightarrow\text{x} \in \text{A and x}\not\in\text{B or x}\in \text{A and x} \in\text{B}$
$\Rightarrow \text{x} \in \text{A}$
$\therefore\text{(A} - \text{B)}\cup\text{(A} \cap \text{B}) \subset \text{A}.....\text{(ii)}$
From (i) and (ii), we get
$\text{(A} - \text{B)}\cup\text{(A} \cap \text{B) = A}.$

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