Posted by Anna on Thursday, October 7, 2010 at 9:56pm.
Use proof by contraposition to prove the following statement:
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 11:12pm
NOTE: 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 ...
 Algebra 2 Could use help asap  I really am trying to work these problems out on...
 math proof  how can i prove that for any set of real numbers, there is at least...
 math  describe the solutions of 4<or=n+2 mult. choice a all real numbers ...
 discrete math  use a direct proof to show that the product of two odd numbers ...
 Algebra 1 Please, check my answers!  1. What is ((12x^6 + x)) / ((4x^2))? A...
 algebra  the equation is f(x) = (1/sq. root of 1x^2) 1. f(x) is never zero. 2...
 math: real numbers  Determine whether the following statement is true or fase. ...
 math 2  let f and g be two functions defined by: f(x)=x+5, with domain all real...
 sorry left out a "x"  If f(x)=3^2 g(x)= 1/6+x find the following and give the ...
More Related Questions