Example of a One-One but Not Onto Function

📺 Video Explanation

📝 Question

Give an example of a function which is:

(i) one-one but not onto.

---

✅ Solution

Consider the function:

\[ f:\mathbb{N}\to\mathbb{N} \]

defined by:

\[ f(x)=x+1 \]

---

🔹 Check One-One (Injective)

Assume:

\[ f(x_1)=f(x_2) \]

Then:

\[ x_1+1=x_2+1 \]

So:

\[ x_1=x_2 \]

✔ Therefore, function is one-one.

---

🔹 Check Onto (Surjective)

For onto, every element in codomain must have a pre-image.

But:

\[ 1\in\mathbb{N} \]

There is no natural number \(x\) such that:

\[ x+1=1 \]

because:

\[ x=0 \]

and \(0\notin\mathbb{N}\) (school convention).

❌ So function is not onto.

---

🎯 Final Answer

An example is:

\[ \boxed{f(x)=x+1,\quad f:\mathbb{N}\to\mathbb{N}} \]

This function is one-one but not onto.

---

🚀 Exam Shortcut

• \(f(x)=x+1\) on natural numbers is a standard example
• Shifting by 1 keeps uniqueness
• First number in codomain is missed → not onto