Proving x ≠ -1/y When x*y ≠ -1

[wfc]

How could you prove that if x*y ≠ -1, then x/y ≠ -1?

x*y ≠ -1 → x ≠ -1/y

I'm not sure where to go after that.

 

[wabbit]

You don't prove it, because it isn't true. Can you find a couterexample ?

 

[micromass]

Assuming it were true (which it isn't), the best way would be to do contraposition. It would then suffices to show that $x/y = -1$ implies $xy = -1$.

 

[wfc]

Can you explain why it isn't true? I'm still confused.

And where you say x/y = -1, then xy = -1, I can't come up with an example where that would work?

 

[micromass]

If you can't come up with an example where that would work, then that means it's false.

 

[wabbit]

What you need to find is an example of $x,y$ such that $x/y=-1$ and $xy\neq -1$ .

Once you've done that you've proved that your initial statement $(xy\neq -1\Rightarrow x/y\neq -1 )$ is false

 

[FL0R1]

How could you prove that if x*y ≠ -1, then x/y ≠ -1?

x*y ≠ -1 → x ≠ -1/y

I'm not sure where to go after that.

if x ≠ -1/y then (-1/y)/y ≠ -1 -> here we go that -1 ≠ -1 so its not true

 

[wabbit]

if x ≠ -1/y then (-1/y)/y ≠ -1

No, this does not follow at all.

 

[Svein]

How could you prove that if x*y ≠ -1, then x/y ≠ -1?

Counterexample:

Put x = -2, y=2. Then x⋅y = -4 (which is not -1) and x/y = -1.

 

[wabbit]

x⋅y = -4 (which is not -1)

You don't leave any stone unturned :)

 

[Svein]

You don't leave any stone unturned :)

I am a mathematician. I have to turn them.

 

[jaskamiin]

It would probably be easiest to first attempt to find a counterexample. (it's pretty easy, if you let x and y be numbers with the same absolute value but opposite signs)

If that for some reason turns out fruitless, you can attempt a proof. Contradiction is probably the easiest (because it's really saying the same thing as finding a specific counterexample!).

Instead of proving that for every x and y in the universe, xy ≠ -1 ⇒ x/y ≠ -1, the negated sentence is a bit easier to bite into: xy≠ -1 ∧ x/y = -1. Prove that two numbers x and y can't exist to make this true.