https://file.aiquickdraw.com/mathbot/file/1730027280214xzkruv9c.jpg

The assignment requires creating either a mathematical or real-life statement and proving it using either **direct proof** or **indirect proof**.

Here’s an example of each type of proof:

### 1. Direct Proof Example
**Statement**: The sum of two even numbers is always even.

**Proof**:  
Let two even numbers be \( a = 2m \) and \( b = 2n \), where \( m \) and \( n \) are integers.  
The sum of \( a \) and \( b \) is:
\[
a + b = 2m + 2n = 2(m + n)
\]
Since \( m + n \) is an integer, \( 2(m + n) \) is also even. Thus, the sum of two even numbers is always even.

### 2. Indirect Proof Example (Proof by Contradiction)
**Statement**: The square root of 2 is irrational.

**Proof**:
Assume, for contradiction, that \(\sqrt{2}\) is rational. This would mean it can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers with no common factors other than 1 (meaning the fraction is in simplest form).  
Then:
\[
\sqrt{2} = \frac{p}{q} \Rightarrow 2 = \frac{p^2}{q^2} \Rightarrow p^2 = 2q^2
\]
Since \(p^2\) is even (because it equals \(2q^2\)), \(p\) must also be even. Let \(p = 2k\) for some integer \(k\).  
Substitute \(p = 2k\) into the equation:
\[
(2k)^2 = 2q^2 \Rightarrow 4k^2 = 2q^2 \Rightarrow q^2 = 2k^2
\]
This implies \(q^2\) is also even, and hence \(q\) is even.  
Since both \(p\) and \(q\) are even, they have a common factor of 2, contradicting our assumption that \(\frac{p}{q}\) is in simplest form.  
Therefore, \(\sqrt{2}\) must be irrational.

Would you like further clarification or help with another statement?

### Related Questions:
1. What are some common real-life situations where indirect proof is useful?
2. How does a direct proof differ from a proof by induction?
3. Can you provide an example of a geometric statement proved using direct proof?
4. Why is the assumption of the opposite often used in indirect proofs?
5. What are some advantages of using indirect proofs over direct proofs?

**Tip**: In indirect proof, always assume the negation of the statement you want to prove and look for a contradiction.

Make a mathematical or a real-life statement and prove it using direct or indirect proof.

Explore how to prove mathematical or real-life statements using direct and indirect proofs. Learn methods of constructing logical arguments and examples such as proving the sum of two even numbers is even and the irrationality of the square root of 2. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation