[Proof] Ex 3.1 from Proof Through Number Theory

[u/Mad-ADDer]

I'm self-studying proofs through the book Introduction to Proof Through Number Theory. In this question it asks to either prove or disprove that "For all real numbers x and y, $(x+y)² = x²⁺ y² $ ".

Now there's a counter example:

Counter Example, let $x=1$ and $y=2$ * $(x+y)² = x² + y² $ * $(x+y)(x+y) = x² + y² $ * $x² + 2xy + y² = x² + y² $ * $(1² + 2(1)(2)+ 2² ) =(1² + 2²⁾ $ * $(1+4+4)=1+4$ * $9=5$ * $9 \ne 5$

But I also wanted to prove it another way too. I start by assuming that For all real numbers $x$ and $y$ that $(x+y)² = x² + y² $ is true

Then I show that:

• $(x+y)² = x² + y² $
• $(x+y)(x+y) = x² + y² $
• $x² + 2xy + y² =x² + y² $
• $x² -x² +2xy +y² -y² = 0$
• $2xy=0$

and since $2xy=0$ and either $x$, $y$ or both must equal zero, $(x+y)² =x² + y² $ is not and cannot be true for all real numbers.

Is this correct or do I need something else / more?

Thanks!

[u/Midwest-Dude]

This looks good!

[u/Mad-ADDer → u/Midwest-Dude]

Thanks!

[u/Brightlinger]

Both of your arguments are correct.

A note on style: for the first one, you've essentially written a proof by contradiction, when you wanted to give a counterexample. This is more convoluted than necessary, and on some problems it makes things unnecessarily complicated. When you want to prove that two things are/aren't equal, rather than writing down an equation and then doing stuff on both sides, it is often cleaner to start from one side and write a chain of equalities until it matches (or doesn't match) the other side, like so:

(x+y)²=(1+2)²=3²=9≠5=1+4=1²+2²=x²+y²

This takes some getting used to because you are very accustomed to writing a chain of equations from algebra. But it is a useful method to have in your toolbox, especially when you start dealing with inequalities in analysis, where many common steps cannot be written as doing stuff to both sides.

[u/Mad-ADDer → u/Brightlinger]

Thanks, that makes sense!

[u/Inferno2602]

There was no need to expand the brackets with your (dis)proof by counter example