Let A and B be sets. Show that f: A B B A such that f (a, b) = (b… - Vidyadip

Question

Let $A$ and $B$ be sets. Show that $f: A \times B \rightarrow B \times A$ such that $f (a, b) = (b, a)$ is bijective function.

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Answer

$f: A \times B \rightarrow B \times A$ is defined as $f(a, b) = (b, a).$
Let $(\text{a}_1,\text{b}_1),(\text{a}_2,\text{b}_2)\in \text{A}\times\text{B}$ such that $f(a_1, b_1) = f(a_2, b_2)$
$\Rightarrow (b_1, a_1) = (b_2, a_2) \Rightarrow b_1 = b_2$ and $a_1 = a_2 \Rightarrow (a_1, b_1) = (a_2, b_2) $
$\therefore f$ is one-one.
Now, let $(\text{b},\text{a})\in\text{B}\times\text{A}$ be any element. Then, there exists $(\text{a},\text{b})\in\text{A}\times\text{B}$ such that $f(a, b) = (b, a). [$By definition of $f]$
$\therefore$ f is onto. Hence, $f$ is bijective.

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