# Homework Help: Counterexample help

1. Mar 1, 2009

### relyt

Find a counterexample to the statement "For all real numbers u and v, (u + v)^2 is not equal to u2 + v2."

Last edited: Mar 1, 2009
2. Mar 1, 2009

### Hurkyl

Staff Emeritus
What work have you done on the problem? Have you tried anything at all? It's actually harder to find an example than it is a counterexample....

(I assume you meant to write something like $(u + v)^2$ -- the proper way to write that when you can't use a superscript is as (u+v)^2, because what you actually wrote is to multiply by 2, and that's a very different problem. :tongue:)

3. Mar 1, 2009

### relyt

Hey Hurkyl,
I've tried a couple of things, but I know they are not right. Should I post them here anyway :(

4. Mar 1, 2009

### Hurkyl

Staff Emeritus
Definitely. It is forum policy that we won't offer much help until you've shown that you've worked on a problem.... (p.s. wee the edits in my previous post)

5. Mar 1, 2009

### symbolipoint

You want a counterexample for a basic property of Real Numbers? Yes, I know what set of numbers would give a counterexample. It is the set {}.

6. Mar 1, 2009

### relyt

Now I'm really confused

7. Mar 2, 2009

### Gib Z

Lol he's just being slight cruel.

Try substitute some small numbers for u and v, see if your Left hand side is equal to your right hand side.

8. Mar 2, 2009

### HallsofIvy

No, he's just being completely wrong.

To find a counter example to "(x+ y)2 is NOT x2+ y2", you must use x and y so that (x+y)2= x2+ y2.
Presumably you know that (x+ y)2= x2+ 2xy+ y2. That means you must find x and y so that x2+ 2xy+ y2= x2+ y2.

Notice that the squares on both sides cancel. What does that leave you with?

9. Mar 2, 2009

### symbolipoint

From original: "(u + v)^2 is not equal to u2 + v2."

Actually, I misread the original relation. Either that or the "2" was not shown as exponentiation; but as shown on the right-hand side, the "2" are not shown as exponents.

10. Mar 2, 2009

### HallsofIvy

No, I read it as "Find a counterexample to 'for all real numbers x,y it is NOT true that $(u+ v)^2= u^2+ v^2$'" and there is an easy counterexample as I pointed out.

Last edited by a moderator: Mar 7, 2009
11. Mar 2, 2009

### symbolipoint

That is not what was written originally. The 2's on the righthand side were not exponentiated.

12. Mar 6, 2009

### Tobias Funke

The notation isn't the problem. The statement "For all real numbers x,y, (x+y)^2!=2x+2y" is still false. I read it the same as Hurkyl.

13. Mar 7, 2009

### Mentallic

Why so? try $y=-x$ and $y=2-x$

edit: I misread "for all real numbers" as "for what real numbers"