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Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsFunctions3 Marks
Question
If A = {1, 2, 3}, show that a one-one function f : A → A must be onto.
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Answer

A = {1, 2, 3}
Number of elements in A = 3
Number of one-one functions = number of ways of arranging 3 elements = 3! = 6
So, the possible one-one functions can be the following:
  1. {(1, 1), (2, 2), (3, 3)}
  2. {(1, 1), (2, 3), (3, 2)}
  3. {(1, 2 ), (2, 2), (3, 3)}
  4. {(1, 2), (2, 1), (3, 3)}
  5. {(1, 3), (2, 2), (3, 1)}
  6. {(1, 3), (2, 1), (3, 2)}
Here, in each function, range = {1, 2, 3}, which is same as the co-domain.
So, all the functions are onto.

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