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In the proof of 2nd one how g is bijective i understood how the have written but did not understood the concept of writing the last sentence

Let's analyze the proof you're asking about to understand the concept behind the last sentence.

### Overview of the Problem
The second theorem in the notes seems to involve concepts from set theory, particularly related to bijections and infinite sets. The theorem can be roughly stated as:

- **Theorem**: If \( A \) is a set with \( n \) elements (finite), and \( B \) is an infinite set, then the set \( A \cup B \) (union of \( A \) and \( B \)) is also infinite.

### Proof Outline
1. **Given**: 
   - \( A \) is a finite set with \( n \) elements.
   - \( B \) is an infinite set.

2. **Claim**: \( A \cup B \) is infinite.

3. **Construction of a Function**:
   - Define a function \( g: B \rightarrow A \cup B \) which maps each element of \( B \) to itself in \( A \cup B \).
   - If the function \( g \) is bijective, \( A \cup B \) is infinite.

4. **Last Sentence**:
   - The proof concludes that \( g \) is bijective, implying \( A \cup B \) is infinite.

### Understanding the Last Sentence
The concept behind the last sentence revolves around the idea that since \( B \) is infinite, and \( g \) is defined such that it essentially maps elements of \(

Explore the proof of the second theorem involving set theory, bijections, and infinite sets. Gain insights into how the function g is bijective, ensuring the set A union B remains infinite. Suitable for undergraduate students studying advanced mathematics.

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