Question

Given below are two statements:

Assertion (A):
A function \(f:N\to N\), defined as \[ f(x)= \begin{cases} x+1, & \text{if } x \text{ is odd}\\ x-1, & \text{if } x \text{ is even}\\ \end{cases} \] is not surjective.
Reason (R):
A function \(f:X\to Y\) is said to be surjective if every element of \(Y\) is the image of some element of \(X\) under \(f\), i.e., for every \(y\in Y\), there exists \(x\in X\), such that \(f(x)=y\).

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

To check surjectivity, verify whether every element of the codomain is obtained as an output of the function.

Answer 1

The Correct Option is D