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 - oneNow, Let $$(b,a)\epsilon B \times A$$ be any elementThen, 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 bijectiveMathematics

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