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Question

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


Solution

Solution :
$$ f : A \times B \rightarrow B \times A $$ is defined as $$ f (a, b) = (b,a)$$
Let $$ (a_{1}, b_{1}), (a_{2}, b_{2}) $$ & $$ A\times B $$ such that
$$ f(a_{1}, b_1) = f(a_2, b_{2})$$
$$ \Rightarrow (b_1, a_1) = (b_2, a_2)$$
$$ \Rightarrow b_{1} = b_{2} $$ & $$ a_{1} = a_{2}$$
$$ \Rightarrow (a_{1}, b_{1}) = (a_{2}, b_{2})$$
$$ \therefore $$ f is one - one
Now, Let $$(b,a)\epsilon  B \times A $$ be any element
Then, there exists $$ (a,b) \epsilon A \times B $$ such that
$$ f(a,b) = (b,a)... $$ [By difinition of f]
f is outo. Hence f is bijective

1060320_876255_ans_9dbf6e1852094dedb2185040992f6204.png

Mathematics

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