Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSets3 Marks
Question
If A and are sets, then prove thet $\text{A - B},\text{ A}\cap\text{B}$ and $\text{A - B}$ are pair wise disjoint.
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Answer
We need to show that $(\text{A - B})\cap(\text{A}\cap\text{B})=\phi,\ (\text{A}\cap\text{B})\cap(\text{A - B})=\phi$ and $(\text{A - B})\cap\text{B - A}=\phi$ The 3 sets $\text{A} - \text{B}, \text{ A}\cap\text{B}$ and B - A may be represented by venn diagram as follows It is clear from the diagram thet the 3 sets are pairwise disjoint, but we shell give a proff of it. We first show that $(\text{A - B})\cap(\text{A}\cap\text{B})=\phi$ Let $\text{x}\in(\text{A - B})$ $\Rightarrow\text{x}\in\text{A and x}\not\in\text{B}$ [by definition of A - B] $\Rightarrow\text{x}\not\in\text{A}\cap\text{B.}$ This is true for all $\text{x}\in(\text{A - B)}$ Hence $(\text{A - B})\cap(\text{A}\cap\text{B})=\phi$ On a similar lines, it can be seen that $(\text{A}\cap\text{B})\cap(\text{A}-\text{B})=\phi$ Finally, we show that $\text{(A - B)}\cap\text{(A - B)}=\phi$ We have, $\text{A - B}=\{\text{x}\in\text{A : x}\not\in\text{B}\}$ and $\text{B - A}=\{\text{x}\in\text{B : x}\not\in\text{A}\}$ Hence, $(\text{A - B})\cap(\text{B - A})=\phi.$
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