Proving 2xy =< x^2 + y^2 for all real numbers x and y: A Contradictory Approach

In summary, the conversation discusses proving that 2xy is less than or equal to x^2 + y^2 for all real numbers x and y. The attempt at a solution uses a direct proof method, showing that since (x-y)^2 is either a positive integer or zero, 0 is less than or equal to (x-y)^2, which is also less than or equal to x^2-2xy+y^2. This leads to the conclusion that 2xy is less than or equal to x^2 + y^2. However, the negation of this statement should be 2xy > x^2 + y^2.
  • #1
skeeterrr
14
0

Homework Statement



Prove for all real numbers x and y that [tex]2xy =< x^2 + y^2[/tex]

Homework Equations


The Attempt at a Solution



Well, since this is a problem regarding proof, I thought I would start with a contradictory statement like:

2xy >= x^2+y^2



0 >= x^2-2xy+y^2



0 >= (x-y)^2

Since (x-y)^2 is either a positive integer and a zero,

0 =< (x-y)^2



0 =< x^2-2xy+y^2



2xy =< x^2+y^2

Well that's all I can think of... can anyone point out any mistakes or anything? Is there more to it than just this?
 
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  • #2
i think that it looks fine
 
  • #3
What you've done is not a Proof by Contradiction.

The bottom part of your post is indeed qualified as a direct proof. No need for Contradiction. :)

Since (x-y)^2 is either a positive integer and a zero,

0 =< (x-y)^2



0 =< x^2-2xy+y^2



2xy =< x^2+y^2

Well that's all I can think of... can anyone point out any mistakes or anything? Is there more to it than just this?

And btw, the negation of 2xy <= x2 + y2 is not

skeeterrr said:
2xy >= x^2+y^2

Instead, it should be: 2xy > x2 + y2.
 

FAQ: Proving 2xy =< x^2 + y^2 for all real numbers x and y: A Contradictory Approach

What is an inequality proof?

An inequality proof is a mathematical argument that shows the relationship between two quantities, where one quantity is greater than or less than the other. It is used to demonstrate that a particular inequality is true for all possible values of the variables involved.

How do I check if my inequality proof is correct?

To check your inequality proof, you can follow these steps:

  • Simplify both sides of the inequality to make it easier to work with.
  • Use algebraic properties and rules to manipulate the inequality.
  • Substitute different values for the variables to see if the inequality holds true.
  • Check if your proof is valid by making sure each step follows logically from the previous one.

What are some common mistakes to avoid when writing an inequality proof?

Some common mistakes to avoid when writing an inequality proof include:

  • Forgetting to include all necessary steps and assumptions.
  • Incorrectly applying algebraic rules and properties.
  • Misinterpreting the meaning of the inequality symbol.
  • Not considering all possible values of the variables involved.

Can I use a calculator to check my inequality proof?

While a calculator can help you with the calculations involved in your inequality proof, it is not recommended to solely rely on it. It is important to understand the concept and logic behind the inequality and to check your proof manually to ensure its validity.

Are there any resources available to help me check my inequality proof?

Yes, there are various online resources and forums where you can seek help and feedback on your inequality proof. You can also consult with a math tutor or professor for guidance and clarification.

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