What is direct proof and give me an example

### **Direct Proof: Definition**

A **direct proof** is a method of proving a mathematical statement by directly demonstrating that the premises logically lead to the conclusion. In a direct proof, you assume that the given statement (often called the hypothesis) is true, and through logical deductions, show that the conclusion must also be true. This approach is commonly used in proving statements that involve universal quantifiers, such as "for all" statements.

### **Example:**

**Statement to prove:**  
"If \( n \) is an even integer, then \( n^2 \) is also even."

**Direct Proof:**

1. **Assume the hypothesis is true:**  
   Let \( n \) be an even integer. By definition of even numbers, there exists some integer \( k \) such that \( n = 2k \).

2. **Apply this to the expression \( n^2 \):**  
   Now, we calculate \( n^2 \):  
   \[
   n^2 = (2k)^2 = 4k^2 = 2(2k^2)
   \]

3. **Conclude the result:**  
   Since \( n^2 = 2(2k^2) \), it is clear that \( n^2 \) is divisible by 2, which means \( n^2 \) is an even number.

Thus, we have shown that if \( n \) is an even integer, then \( n^2 \) is also even.

This is a **direct proof** because we assumed \( n \) is even and directly showed that this assumption leads to \( n^2 \) being even as well.

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Would you like more details or examples? Here are 5 related questions for further exploration:

1. How does a direct proof differ from a proof by contradiction?
2. Can you provide a direct proof for the statement: "If a number is divisible by 4, it is divisible by 2"?
3. What are some common mistakes to avoid in direct proofs?
4. How would you approach proving "If \( a \) and \( b \) are even, their sum is also even"?
5. Can direct proof be used in all types of mathematical theorems?

**Tip:** Always ensure your definitions and assumptions are clear when starting a proof—this makes the logic easier to follow.

Explore the concept of direct proof with a step-by-step example. Learn how to prove that if n is an even integer, then n^2 is also even, using logical deduction and the definition of even numbers. Suitable for students in Grades 9-12.

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