How Does Proof by Contradiction Validate a/b + b/a >= 2?

In summary, proof by contradiction is a method of proving a mathematical statement by assuming the opposite and showing that it leads to a contradiction, thus proving the original statement must be true. To use this method, one must assume the opposite of the statement and use logical reasoning to reach a contradiction. This method can be used for both existential and universal statements, but has limitations as it can only be used for statements that are logically equivalent to their contrapositives and cannot prove untrue statements or those with counterexamples. Proof by contradiction can also be used in other fields besides math, such as philosophy, computer science, and law.
  • #1
chan8366
3
0
[SOLVED] basic math proof by contradiction

Homework Statement



prove: If a and b are positive numbers, a/b +b/a>=2

Homework Equations





The Attempt at a Solution



by contradiction (a^2+b^2)/ab<2 and got lost
 
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  • #2
If (a^2+b^2)/ab<2 then a^2+b^2<2ab. So a^2+b^2-2ab<0. But a^2+b^2-2ab=(a-b)^2<0. What could be wrong with that?
 
  • #3
thanks it was that easy
 

Related to How Does Proof by Contradiction Validate a/b + b/a >= 2?

1. What is proof by contradiction in basic math?

Proof by contradiction is a method of proving a mathematical statement by assuming the opposite, or contradiction, of the statement and showing that it leads to a contradiction or impossibility. This allows us to conclude that the original statement must be true.

2. How do you use proof by contradiction to prove a statement?

To use proof by contradiction, you first assume the opposite of the statement you want to prove. Then, you use logical reasoning and mathematical principles to show that this assumption leads to a contradiction. This proves that the original statement must be true.

3. What types of statements can be proved using proof by contradiction?

Proof by contradiction can be used to prove both existential and universal statements. An existential statement is a statement that asserts the existence of something, while a universal statement is a statement that applies to all elements in a set.

4. Are there any limitations to using proof by contradiction?

Proof by contradiction can only be used to prove statements that are logically equivalent to their contrapositives. This means that the statement must be in the form "if A, then B" and the contrapositive is "if not B, then not A". Additionally, proof by contradiction cannot be used to prove statements that are not true or have counterexamples.

5. Can proof by contradiction be used in other fields besides math?

Yes, proof by contradiction can be used in other fields such as philosophy, computer science, and even law. It is a powerful tool for proving statements and arguments in various disciplines that require logical reasoning and deductive thinking.

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