Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSETS1 Mark
Question
Show that for any sets A and B, A = ( A $\cap$B ) $\cup$( A – B ) and A $\cup$( B – A ) = ( A $\cup$B )
✓
Answer
We have to prove: $A=(A \cap B) \cup(A-B)$ Proof: Let $x \in A$ Now, we need to show that $x \in(A \cap B) \cup(A-B)$ In Case I, $x \in(A \cap B)$ $\Rightarrow X \in(A \cap B) \subset(A \cup B) \cup(A-B)$ In Case II, $X \notin A \cap B$ $\Rightarrow x \notin B \ or \ x \notin A$ $\Rightarrow X \notin B(X \notin A)$ $\Rightarrow X \notin A-B \subset(A \cup B) \cup(A-B)$ $\therefore A \subset(A \cap B) \cup(A-B)(i)$ It can be concluded that, $A \cap B \subset A \ and \ (A-B) \subset A$ Therefore, (A $\cap$B) $\cap$(A - B) $\subset $A (ii) Equating (i) and (ii),we get $A=(A \cap B) \cup(A-B)$ Now, we need to show, $A \cup(B-A) \subset A \cup B$ Suppose that, $X \in A \cup(B-A)$ $X \in A \ or \ X \in(B-A)$ $\Rightarrow X \in A \ or \ (X \in B \text { and } X \notin A)$ $\Rightarrow(X \in A \text { or } X \in B) \ and \ (X \in A \text { and } X \notin A)$ $\Rightarrow X \in(B \cup A)$ $\therefore A \cup(B-A) \subset(A \cup B)$(iii) Now, to prove: $(A \cup B) \subset A \cup(B-A)$ Let y $\in A\cup B$ $\mathrm{y} \in \mathrm{A}\ or \ \mathrm{y} \in \mathrm{B}$ $(y \in A \text { or } y \in B)\ and \ (X \in A \text { and } X \notin A)$ $\Rightarrow y \in A \ or \ (y \in B \text { and } y \notin A)$ $\Rightarrow y \in A \cup(B-A)$ Therefore,$A \cup B \subset A \cup(B-A)$(iv) $\therefore$ Using (iii) and (iv), we obtain $A \cup(B-A)=A \cup B$
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