Let A be the set of all 50 students of Class X in a school. Let $f: \mathrm{A} \rightarrow \mathbf{N}$ be function defined by $f(x)=$ roll number of the student $x$. Show that $f$ is one-one but not onto.

Step 1: Understand the Function Definition

We are given a set $A$ containing 50 students from Class X. The function $f$ maps each student in set $A$ to their unique roll number, which is a natural number. The codomain of the function is the set of natural numbers, $\mathbf{N} = \{1, 2, 3, \dots\}$.

Step 2: Prove One-One (Injective)

To prove that a function is one-one, we assume that $f(x_1) = f(x_2)$ for two elements $x_1, x_2$ in the domain $A$. If this assumption leads to $x_1 = x_2$, then the function is one-one. In this case, if two students have the same roll number, they must be the same student, as roll numbers are unique identifiers.

Step 3: Prove Not Onto (Surjective)

For a function to be onto, every element in the codomain must have at least one pre-image in the domain. Here, the domain $A$ has 50 elements, so the function can map to at most 50 distinct roll numbers. However, the codomain $\mathbf{N}$ (natural numbers) is infinite. This means there are infinitely many natural numbers that are not roll numbers of any student in Class X, and thus have no pre-image in $A$.

Step 4: Conclusion

We have successfully demonstrated that the function $f$ is one-one because each student has a unique roll number. We have also shown that $f$ is not onto because the set of natural numbers (codomain) contains elements that are not roll numbers of any student in the given class.

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