I really do not get proofs AT ALL.

[XodoX]

I really do not get proofs AT ALL. Stuff like this...

"Prove that (n+1)^{2 $$\geq$$}3ⁿ if n is a positive integer with n$$\leq$$4."

Proof by exhaustion would be applied here.. what the book tells me.




"Show that there are no solutions in integers x and y of x²+3y²=8."

Then there's also "Constructive Existence Proof" and "Nonconstructive proof". No idea how to do anyone of them. How do I approach those equations, and how do I know which proof I need to use??

 

[EnumaElish]

For the first problem, "by exhaustion" means "case by case."

(Non-) Constructive proof: http://en.wikipedia.org/wiki/Constructive_proof

 

[Hurkyl]

I've edited out some whining and language. :grumpy:

 

[Hurkyl]

Something is a (formal) proof if and only if it adheres to a certain set of directions. (the rules of logic)

There is a wide variety of proofs, because the rules of logic allow many different possibilities at each step of the proof.

The "game" of proving things is to find a proof that starts with a useful hypothesis and ends with a useful conclusion. (And in this problem, you're told what the hypothesis and the conclusion are)



Now, what do the rules of logic say a "proof by exhaustion" is?

 

[blue_raver22]

As a scientist, it is important to understand the concept of proof in order to accurately and confidently support any hypotheses or theories. Proofs are essential in the field of science as they provide evidence and support for claims or statements. They allow us to validate our ideas and make logical conclusions based on established facts and principles.

Proofs can be intimidating and confusing, especially if you are not familiar with mathematical notation or terminology. However, with practice and understanding, anyone can learn to read and write proofs effectively.

In regards to the first statement, it is important to understand that proofs are a way of logically showing that a statement is true. In this case, the statement is (n+1)2 \geq3n if n is a positive integer with n\leq4. The use of proof by exhaustion means that we will consider all possible values of n (1, 2, 3, and 4) and show that the statement holds true for each of those values. This is a valid approach for this particular problem because there are only a finite number of possible values for n.

For the second statement, x2+3y2=8, we can use a nonconstructive proof. This type of proof shows that a solution does not exist by assuming that a solution exists and then arriving at a contradiction. In this case, we can assume that there are integers x and y that satisfy the equation, and then show that this leads to a contradiction. This would prove that no solutions exist.

As for which proof to use, it is important to first understand the problem and the information given. In some cases, the type of proof may be specified or hinted at in the problem itself. In other cases, you may need to use your understanding of the problem and your knowledge of different proof methods to determine the best approach.

In conclusion, proofs are an important tool in the scientific method and understanding them is crucial for any scientist. With practice and familiarity, you can develop the skills needed to read and write proofs effectively. It is also important to understand the different types of proofs and when to use them in order to approach problems effectively.