https://file.aiquickdraw.com/mathbot/file/1729825206268mjtjn1hi.jpg

Let's go through the image you uploaded. The task asks to fill in the blanks of two proof exercises, then identify the type of proof. I'll guide you through the solutions:

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### a) **Prove that if \( n \) is an odd integer, then \( n^2 \) is odd.**

**Proof:**

- Suppose that **\( n \) is an odd integer**. Then, \( n = 2k + 1 \), for some integer \( k \).

Now, 
\[ 
n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1 
\]
where \( s = 2k^2 + 2k \). Since \( s \) is an integer, then \( n^2 \) is **odd** when \( n \) is odd. **QED**

**Type of Proof**: **Direct Proof**

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### b) **Prove that if \( x^2 \) is an even integer, then \( x \) is even.**

**Proof:**

- Suppose \( x^2 \) is even and **\( x \) is not even**. Since \( x \) is not even, then \( x \) is odd. Hence, by definition of an odd integer, \( x = 2k + 1 \), for some integer \( k \).

Now,
\[ 
x^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1 
\]
where \( s = 2k^2 + 2k \). Since \( s \) is an integer, then \( x^2 \) is **odd**. This is a **contradiction** to the assumption that \( x^2 \) is even.

**Type of Proof**: **Proof by contradiction**

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### Based on Activity 1:

#### 2.1 How did you complete the proof?

The proof was completed by substituting the expressions for odd and even integers and simplifying the results, then verifying whether the assumption holds or leads to a contradiction.

#### 2.2 Is it important to know the basic concepts or theorems in writing proof? Why or why not?

Yes, it is important because basic concepts and theorems provide a foundation for logical reasoning, ensuring that each step in the proof is valid and leads to the correct conclusion.

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Would you like any further details or clarifications?

### 5 Related Questions:
1. What is the difference between direct proof and proof by contradiction?
2. Can you provide an example where \( n^2 \) is odd but \( n \) is not an odd integer?
3. How does assuming the opposite in a proof by contradiction help establish the original statement?
4. Why is it necessary to show that \( s \) is an integer in these proofs?
5. How would you approach proving that if \( x^2 \) is odd, then \( x \) is odd?

### Tip:
Always ensure that your assumptions in a proof align with the definitions of the mathematical concepts you're working with, like odd or even integers.

Fill in the missing parts to complete the proof and identify the methods used in the given mathematical exercises.

Solve mathematical proof problems involving odd and even integers using direct proof and proof by contradiction techniques. This problem explores integer properties and key proof strategies suitable for high school students (Grades 9-12).

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