Counterexample: (u + v)^2 ≠ u2 + v2

[relyt]

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

 

[Hurkyl]

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. )

 

[relyt]

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

 

[Hurkyl]

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

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)

 

[symbolipoint]

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

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 {}.

 

[relyt]

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 {}.

Now I'm really confused

 

[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.

 

[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?

 

[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.

 

[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.

 

[symbolipoint]

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

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

 

[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.

 

[Mentallic]

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.

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

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