CBSE BoardEnglish MediumSTD 11 ScienceMathsModel Paper 72 Marks
Question
If $A$ and $B$ are any two non-empty sets, then prove that $A \times B=B \times A \Leftrightarrow A=B$
✓
Answer
First, let $A = B$. Then, we have to prove that $A \times B=B \times A$
Now, $A = B$
$\Rightarrow A \times B=A \times A \text { and } B \times A=A \times A$
$\Rightarrow A \times B=B \times A$
Conversely, let $A \times B=B \times A$. Then, we have to prove that $A = B$.
Let x be an arbitrary element of A. Then,
$x \in A$
$\Rightarrow (x, b) \in A \times B \text { for all } b \in B .$
$\Rightarrow (x, b) \in B \times A$
$\Rightarrow x \in B$
$\therefore A \subseteq B$
Let y be an arbitrary element of A. Then,
$y \in B$
$\Rightarrow (a, y) \in A \times B \text { for all } a \in A$
$\Rightarrow (a, y) \in B \times A$
$\Rightarrow y \in A$
$\therefore B \subseteq A$
Hence, A = B.
Hence, $A \times B=B \times A \Leftrightarrow A=B$
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