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Relation — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSRelation4 Marks
Question
If $\text{A}\times\text{b}\subseteq\text{C}\times\text{D and A}\times\text{B}=\phi,$ prove that $\text{A}\subseteq\text{C and B}\subseteq\text{D}$
✓

Answer

Let (a, b) be an arbitrary element of A × B. then, $(\text{a},\text{b})\in\text{A}\times\text{B}$ $\Rightarrow\text{a}\in\text{A}\text{ and b}\in\text{B}\ ...(\text{i})$ Now, $(\text{a, b})\in\text{A}\times\text{B}$ $\Rightarrow(\text{a},\text{ b})\in\text{C}\times\text{D}$ $\big[\because\text{ A}\times\text{B}\subseteq\text{C}\times\text{D}\big]$ $\Rightarrow\text{a}\in\text{C and b}\in\text{D}\ ...(\text{ii})$ $\therefore\ \text{a}\in\text{A}$ $\Rightarrow\text{a}\in\text{C}$ [Using (i) and (ii)] $\Rightarrow\text{A}\subseteq\text{C}$ and, $\text{b}\in\text{B}$ $\Rightarrow\text{b}\in\text{D}$ $\Rightarrow\text{B}\subseteq\text{D}$ Hence, proved.

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