dx said:No, those arn't consecutive even integers. Just put x = 1, you get 2 and 3. 3 is not even.
HallsofIvy said:Ok, "two consectutive even numbers" can be represented as 2x and 2x+ 2. Now, what is the sum of their squares? What is the remainder of that number when divided by 8? And why did you label this "attempt at proof by contradiction" when there was no such attempt? And since the problem says "prove algebraically" I see no reason to even try proof by contradiction.
An attempt at proof by contradiction is a mathematical proof technique that involves assuming the opposite of what we are trying to prove and then showing that this leads to a contradiction. If we can prove that the opposite leads to a contradiction, then we can conclude that our original statement is true.
Verification is necessary in an attempt at proof by contradiction because there is always a possibility that our initial assumption or reasoning may be incorrect. By verifying our proof, we can ensure that our logic is sound and our conclusion is valid.
The key steps in an attempt at proof by contradiction are: 1) assuming the opposite of what we are trying to prove, 2) using this assumption to derive a contradiction, and 3) concluding that our original statement must be true because the opposite leads to a contradiction.
No, proof by contradiction is not always a valid method of proof. It can only be used in certain situations where the statement we are trying to prove has a clear opposite or negation. Additionally, the steps must be followed carefully and the contradiction must be logically sound for the proof to be valid.
Yes, there are alternative methods of proof such as direct proof, proof by induction, and proof by contrapositive. Each method may be more suitable for different types of proofs and it is important for a scientist to be familiar with multiple proof techniques.