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.
Domain: positive real numbers
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
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)
≡ ¬(x≥A ∨ y≥A)
≡ x<A ∧ y<A [de Morgan]
⇒ (x+y)/2 < A
⇒ (x+y)/2 ≠ A [since A=(x+y)/2]
Therefore we have proved ¬Q(x,y)⇒¬P(x,y) QED
You can reformulate your answer along the above lines.
Answer this Question
More 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...
- math - describe the solutions of 4<or=n+2 mult. choice a all real numbers ...
- 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 - Working with simple addition and multiplication of positive and negative ...
- Algebra 2 Could use help asap - I really am trying to work these problems out ...
- Math! Please check! - Consider the statement: If two whole numbers are even, ...
- algebra - true or false 1. fraction cant be written as decimal. 2. natural ...