Show that an onto function f : 1, 2, 3 1, 2, 3 is always one-one. - Vidyadip

Question

Show that an onto function f : {1, 2, 3} $\rightarrow$ {1, 2, 3} is always one-one.

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Answer

Suppose f is not one-one.
Then there exists two elements, say 1 and 2 in the domain whose image in the co-domain is the same.
Also, the image of 3 under f can be only one element.
Therefore, the range set can have at the most two elements of the co-domain {1, 2, 3}, showing that f is not onto, a contradiction. Hence, f must be one-one.

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