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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.


Solution

$$f: A \times B \rightarrow B \times A$$ is defined as $$f(a,b)=(b,a)$$.

Let $$(a_1, b_1), (a_2, b_2) \in A \times B$$ such that $$f(a_1, b_1)= f(a_2, b_2)$$.
$$\Rightarrow (b_1, a_1)=(b_2, a_2)$$
$$\Rightarrow b_1 = b_2)$$ and $$(a_1 = a_2)$$
$$\Rightarrow (a_1, b_1) = (a_2, b_2)$$
$$\therefore f$$ is one-one.
Now, let $$(b,a)\in B \times A$$ be any element.
Then, there exists $$(a,b)\in A \times B$$ such that

$$f(a,b)=(b,a)$$. $$[$$By definition of $$f]$$

$$\therefore f$$ is onto.
Hence, $$f$$ is bijective.

Mathematics
RS Agarwal
Standard XII

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