Q237 / 269 · NDA 2025 (Phase 1) · Mathematics
Consider the following for the two (02) items that follow:
Let $f = \{(1, 1), (2, 4), (3, 7), (4, 10)\}$.
Consider the following statements:
I. $f$ is one-one function.
II. $f$ is onto function if the codomain is the set of natural numbers.
Which of the statements given above is/are correct?
Let $f = \{(1, 1), (2, 4), (3, 7), (4, 10)\}$.
Consider the following statements:
I. $f$ is one-one function.
II. $f$ is onto function if the codomain is the set of natural numbers.
Which of the statements given above is/are correct?
- Correct answerAI only
- BII only
- CBoth I and II
- DNeither I nor II
Solution
The given function is $f = \{(1, 1), (2, 4), (3, 7), (4, 10)\}$.
The domain of $f$ is $\{1, 2, 3, 4\}$ and the range of $f$ is $\{1, 4, 7, 10\}$.
Since each element in the domain has a distinct image in the range, $f$ is a one-one function. Statement I is correct.
If the codomain is the set of natural numbers, the range $\{1, 4, 7, 10\}$ is a proper subset of the codomain. Since the range is not equal to the codomain, $f$ is not an onto function. Statement II is incorrect.
Therefore, only Statement I is correct.
Answer: I only
The domain of $f$ is $\{1, 2, 3, 4\}$ and the range of $f$ is $\{1, 4, 7, 10\}$.
Since each element in the domain has a distinct image in the range, $f$ is a one-one function. Statement I is correct.
If the codomain is the set of natural numbers, the range $\{1, 4, 7, 10\}$ is a proper subset of the codomain. Since the range is not equal to the codomain, $f$ is not an onto function. Statement II is incorrect.
Therefore, only Statement I is correct.
Answer: I only
Hence, the correct answer is option: 'A'.