# Discreet Mathematical Structures

posted by .

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 -

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

## Similar Questions

1. ### discrete math

use a direct proof to show that the product of two odd numbers is odd. Proofs: (all the nos. i used are odd) 3 x 3 = 9 5 x 9 = 45 7 x 3 = 21 Yes, but you didn't prove the statement for "all" odd integers, only the odd integers you …
2. ### math

Prove that at least one of the real numbers A1, A2, …, A(n) is greater than or equal to the average of these numbers. What kind of proof did you use?
3. ### math proof

how can i prove that for any set of real numbers, there is at least one number in the set that is greater or equal to the average of the set.
4. ### Algebra II

In an induction proof of the statement 4+7+10+...+(3n-1)=n(3n+5)/2 the first step is to show that the statement is true for some integers n. Note:3(1)+1=1[3(1)+5]/2 is true. Select the steps required to complete the proof. A)Show that …
5. ### Math

Working with simple addition and multiplication of positive and negative numbers. Have completed 59 of 60 problems on a worksheet, I am stumped by one of them. For example Two numbers are given 30 and 13, You must find two number multiplied …
6. ### geometry

1)given-AB=CD,CD=EF prove-AB-EF (in 5 steps that show why it is corect; i.e two-column proof) 2)Given-<2=<3, <4=<5 prove-<1 is supplementay to <6 (in 3 steps, with the two column proof) TWO-COLUMN PROOF LOOKS LIKE …
7. ### algebra

the equation is f(x) = (1/sq. root of 1-x^2) 1. f(x) is never zero. 2. 0 is in the domain of f 3. All negative real numbers are in the domain of f 4. All positive real numbers are in the domain of f 5. 1 is in the domain of f 6. f(x) …
8. ### Algebra 2 Could use help asap

I really am trying to work these problems out on my own. Please help If f(x)=3x^2 g(x)= 1/6+x find the following and give the domain (f+g) (x) (3x^2) + (1/6+x) x is all real numbers and x is not equal to 6 (f-g) (x) (3x^2) - (1/6+x) …
9. ### Algebra 2

A function is created to represent the costs of living per person in the family. What restrictions would be made to the domain?
10. ### Maths

Use proof by contraposition to prove that the following statement is true for all positive integers n: If n2 is a multiple of 3, then n is a multiple of 3.

More Similar Questions