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Relations — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsRelations3 Marks
Question
Let A and B be two sets. Show that the sets A × B and B × A have elements in common iff the sets A and B have an elements in common.
✓

Answer

Let (a, b) be an arbitrary element of $(\text{A}\times\text{B})\cap(\text{B}\times\text{A}).$ Then,
$(\text{a},\text{b})\in(\text{A}\times\text{B})\cap(\text{B}\times\text{A})$
$\Leftrightarrow(\text{a},\text{b})\in\text{A}\times\text{B}$ and $(\text{a},\text{b})\in\text{B}\times\text{A}$
$\Leftrightarrow(\text{a}\in\text{A}\text{ and b}\in\text{B})$ and $(\text{a}\in\text{B and b}\in\text{A})$
$\Leftrightarrow(\text{a}\in\text{A and a}\in\text{B})$ and $(\text{b}\in\text{A and b}\in\text{B})$
$\Leftrightarrow\text{a}\in\text{A}\cap\text{B}$ and $\text{b}\in\text{A}\cap\text{B}$
Hence, the sets A × B and B × A have an element in comon iff the sets A and B have an element in common.

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