Kindly solve this question based on Functions ?
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
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