1. ## Mathematical proofs

For all real numbers x and y, $sqrt(xy)$ 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

2. Originally Posted by treetheta
For all real numbers x and y, $sqrt(xy)$ 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, $\sqrt{xy} \leq \dfrac{x+y}{2}$?

FALSE: take $x=y=-1$

3. Start with this: $(x-y)^2 \geq 0 \qquad \forall x, y \in \mathbb{R}^{\color{red}+}$

Playing around with it gives us:
\begin{aligned}x^2 - 2xy + y^2 & \geq 0 \\ x^2 + 2xy + y^2 & \geq 4xy \\ (x+y)^2 & \geq 4xy \\ & \ \ \vdots\end{aligned}

4. 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 $n^2$ is an odd integer, then n is an odd integer)

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

5. 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 $n^2$ is an odd integer, then n is an odd integer)

would the contra positive be replacing the odd's with even?
No!

Let p = " $n^2$ 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."

6. so the contrapositive would be if n is even then n^2 is even?