Let A and B be sets- Show that f - A - B - B - A such that f a - b - b - a is bijective function-

Here, f: A × B → B × A is defined as f(a,b)=(b,a) Let (a_1, b_1),(a_2,b_2) ∈ A × B Such that f(a_1,b_1)=f(a_2,b_2)⇒ (a_1,b_1)=(a_2,b_2) Therefore, f is one one. ...

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Standard XII

Mathematics

Difference of Two Sets

Let A and B b...

Question

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

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Solution

Here, $f : A \times B \to B \times A$ is defined as f(a,b)=(b,a)
Let $(a_{1}, b_{1}), (a_{2}, b_{2}) \in A \times B$
Such that $f (a_{1}, b_{1}) = f (a_{2}, b_{2}) \Rightarrow (a_{1}, b_{1}) = (a_{2}, b_{2})$
Therefore, f is one-one. Now, let $(b, a) \in B \times A$ of any element.
Then, there exist $(a, b) \in A \times B$ such that f(a,b)=(b,a)[definition of f]
Therefore, f is onto. Hence, f is bijective.

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Let A and B be sets. Show that f:A×B→B×A such that f(a,b)=(b,a) is bijective function.

Here, f:A×B→B×A is defined as f(a,b)=(b,a) Let (a1,b1),(a2,b2)∈A×B Such that f(a1,b1)=f(a2,b2)⇒(a1,b1)=(a2,b2) Therefore, f is one-one. Now, let (b,a)∈B×A of any element. Then, there exist (a,b)∈A×B such that f(a,b)=(b,a)[definition of f] Therefore, f is onto. Hence, f is bijective.

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