# Kindly solve this question based on Functions ?

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Which of the following statement(s) is(are) correct, Explain with some example ?

(A) If #f# is a one-one mapping from set A to A , then #f# is onto.

(B) If #f# is an onto mapping from set A to A , then #f# is one-one.

Which of the following statement(s) is(are) correct, Explain with some example ?

(A) If

(B) If

##### 1 Answer

#### Answer:

This is not true for infinite sets.

#### Explanation:

Counterexample 1:

Let

Then f is one-to-one (with its inverse being the natural logarithm), but f is not onto; its range is the positive numbers.

Counterexample 2:

Let f be defined on the natural numbers as follows:

f(1) = 1.

For n > 1, f(n) = n - 1.

Then f(2) = f(1), so f is not one-to-one.

However, every natural number is in the image of the function, so f is onto.

For finite sets it is true that f is one-to-one if and only if f is onto.

Let | A | be the cardinality of the finite set, A, and let |f(A)| be the cardinality of the image of A under f.

Assume f is one-to-one. Then | A | = | f(A) |, by the definition of one-to-one. Since A is finite and

Assume instead that f is onto. Then for each a