Problem 8 Solution

This page contains the NCERT mathematics class 12 chapter Relations and Functions Exercise 1.2 Problem 8 Solution. Solutions for other problems are available at Exercise 1.2 Solutions
Exercise 1.2 Problem 8 Solution
8. Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.
To check whether f is one-one
In this function, the domain is A × B and the co-domain is B × A
Also note that every element in the domain A × B will be in the form {(a, b)} i.e. it not be a single element but it will be an ordered pair.
Now, the function f is defined as {f(a, b) = (b, a)}
Let (a_1, b_1), (a_2, b_2)A × B be two elements in the domain such that
{f(a_1, b_1) = f(a_2, b_2)}
{(b_1, a_1) = (b_2, a_2)} (∵ f is defined as {f(a, b) = (b, a)})
f is one-one.
To check whether f is onto.
As f is defined as {f(a, b) = (b, a)}, it can be interpreted that every element (b, a)B × A in the co-domain will be an image of the element (a, b)A × B in the domain.
f is onto.
As f is both one-one and onto, f will be bijective.