Thursday

December 18, 2014

December 18, 2014

Posted by **Anna** on Thursday, October 7, 2010 at 9:56pm.

If A is the average of two positive real numbers then one of the two numbers is greater than or equal to A.

Proof.

Domain: positive real numbers

P: A=(x+y)/2

Q: x is greater than or equal to A or y is greater than or equal to A

Method: Assume P is true and Q is false and find a contradiction.

- Discret Mathematical Structures -
**MathMate**, Thursday, October 7, 2010 at 11:12pmNOTE: Contrapositive is not the same as contradiction.

Given: p → q

If we would like to prove by*contrapositive*, we need to prove:

¬q → ¬p

since

p → q ≡ ¬q → ¬p

In the given case,

P(x,y) : "A is the average of two real numbers x and y"

Q(x,y) : (x≥A)∨(y≥A)

We will attempt to prove that:*∀x,y∈ℝ ¬Q(x,y) → ¬P(x,y)*

¬Q(x,y)

≡ ¬(x≥A ∨ y≥A)

≡ x<A ∧ y<A [de Morgan]

So

¬Q(x,y)

⇒ (x+y)/2 < A

⇒ (x+y)/2 ≠ A [since A=(x+y)/2]

⇒ ¬P(x,y)

Therefore we have proved ¬Q(x,y)⇒¬P(x,y) QED

You can reformulate your answer along the above lines.

**Answer this Question**

**Related Questions**

math - Prove that at least one of the real numbers A1, A2, …, A(n) is greater ...

discrete math - use a direct proof to show that the product of two odd numbers ...

math proof - how can i prove that for any set of real numbers, there is at least...

p - Alice and Bob each choose at random a real number between zero and one. We ...

probability - Alice and Bob each choose at random a real number between zero and...

math - describe the solutions of 4<or=n+2 mult. choice a all real numbers ...

algebra - true or false 1. fraction cant be written as decimal. 2. natural ...

Math - The average of 3 numbers is 45. If one of the numbers is 43, what is the ...

finding numbers - The sum of two positive numbers is 20. Find the numbers if the...

math: real numbers - Determine whether the following statement is true or fase. ...