# Mathematical proofs

• Jan 30th 2009, 06:29 AM
treetheta
Mathematical proofs
For all real numbers x and y, $\displaystyle sqrt(xy)$ is less than or equal to

X + Y
-----
2

provide a counter example for statements that are false and provide a complete proof for those that are true
• Jan 30th 2009, 06:52 AM
kalagota
Quote:

Originally Posted by treetheta
For all real numbers x and y, $\displaystyle sqrt(xy)$ is less than or equal to

X + Y
-----
2

provide a counter example for statements that are false and provide a complete proof for those that are true

you mean, $\displaystyle \sqrt{xy} \leq \dfrac{x+y}{2}$?

FALSE: take $\displaystyle x=y=-1$
• Jan 30th 2009, 06:54 AM
o_O
Start with this: $\displaystyle (x-y)^2 \geq 0 \qquad \forall x, y \in \mathbb{R}^{\color{red}+}$

Playing around with it gives us:
\displaystyle \begin{aligned}x^2 - 2xy + y^2 & \geq 0 \\ x^2 + 2xy + y^2 & \geq 4xy \\ (x+y)^2 & \geq 4xy \\ & \ \ \vdots\end{aligned}
• Jan 30th 2009, 06:56 AM
treetheta
oh wow thanks
I was thinking so hard on that one too,

can you help me out with something else too i got a test in 2 hours xD

If we are asked to write the contrapostive of a conditional statement ( For each integer n, if $\displaystyle n^2$ is an odd integer, then n is an odd integer)

would the contra positive be replacing the odd's with even?
• Jan 30th 2009, 07:00 AM
Chop Suey
Quote:

Originally Posted by treetheta
oh wow thanks
I was thinking so hard on that one too,

can you help me out with something else too i got a test in 2 hours xD

If we are asked to write the contrapostive of a conditional statement ( For each integer n, if $\displaystyle n^2$ is an odd integer, then n is an odd integer)

would the contra positive be replacing the odd's with even?

No!

Let p = "$\displaystyle n^2$ is an odd integer" and q = "n is an odd integer"

The contrapositive of "If p, then q" is: "If ~q, then ~p."

The negation of p is "n^2 is an even integer", and the negation of q is "n is an even integer."
• Jan 30th 2009, 07:08 AM
treetheta
so the contrapositive would be if n is even then n^2 is even?