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Question

Let S={1,2,3}. Determine whether the functions f:SS defined as below has inverse. Find f1, if it exists.
(i) f={(1,1),(2,2),(3,3)}
(ii) f={(1,2),(2,1),(3,1)}
(iii) f={(1,3),(3,2),(2,1)}

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Solution

Checking for one-one.
f={(1,1),(2,2),(3,3)}
Since each element has distinct image, f is one-one.

Checking for onto.
f={(1,1),(2,2),(3,3)}
Since, for every image, there is a corresponding element,
f is onto.

Finding inverse of f.
f={(1,1),(2,2),(3,3)} is both one-one and onto. So f is invertible
f1={(1,1),(2,2),(3,3)}

f={(1,2),(2,1),(3,1)}
Since 2 & 3 have the same image 1.
f is not one-one.
Since, f is not -ne-one, it's inverse does not exist.

Checking for one-one.
f={(1,3),(3,2),(2,1)}
Each element has distinct image
f is one-one.

Checking for onto.
f={(1,3),(3,2),(2,1)}
For every image, there is a corresponding element.
f is onto.

Finding inverse of f
Hence, f1={(1,3),(3,2),(2,1)}

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