Math Problem Statement
The Foundations: Logic and Proofs create a example of logic proof
Solution
Example of a Logic Proof: Proof by Contradiction
Problem Statement:
Prove that the square root of 2 is irrational, i.e., cannot be expressed as a ratio of two integers.
Proof:
We'll use a proof by contradiction. The idea is to assume the opposite of what we want to prove and then show that this assumption leads to a contradiction.
-
Assumption (Opposite of What We Want to Prove):
Assume is rational. This means that can be expressed as a fraction: where and are integers with no common factors (other than 1), and . -
Square Both Sides:
Squaring both sides of the equation gives: Multiply both sides by to eliminate the fraction: -
Implication for :
This equation shows that is even (since it equals 2 times something). If is even, then must also be even (because the square of an odd number is odd). -
Express as an Even Number:
Since is even, we can write for some integer . -
Substitute into the Equation:
Substitute into the equation : Simplifying, we get: This equation shows that is even, which means is also even. -
Contradiction: Now we have shown that both and are even, which means that both and have a common factor of 2. However, this contradicts our original assumption that and have no common factors other than 1.
-
Conclusion: Since our assumption that is rational leads to a contradiction, we must conclude that is irrational.
This concludes the proof.
Would you like more details on any step, or do you have any questions?
Here are 8 related questions:
- What is the difference between proof by contradiction and proof by contrapositive?
- How can you prove that other square roots, like , are irrational?
- Can you provide an example of a direct proof?
- What is the importance of logic in mathematical proofs?
- How do you prove a statement using proof by induction?
- What are the key differences between different types of proofs?
- How can you prove that an infinite number of primes exist?
- Can you explain what a constructive proof is?
Tip: When working with proofs, always start by clearly stating your assumptions and what you need to prove. This helps guide your logic and ensures clarity.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Logic
Proofs
Number Theory
Formulas
-
Theorems
Proof by Contradiction
Suitable Grade Level
Advanced High School and above