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.
$(\text{a},\text{b})\in\text{A}\times\text{B}$
$\Rightarrow\text{a}\in\text{A}\text{ and b}\in\text{B}\ ...(\text{i})$
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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