How Do You Prove x ≠ -1/y When x*y ≠ -1?

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Discussion Overview

The discussion revolves around the question of whether it can be proven that if \( x \cdot y \neq -1 \), then \( x/y \neq -1 \). Participants explore the validity of this implication and consider counterexamples and proof strategies.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that if \( x \cdot y \neq -1 \), then \( x \neq -1/y \) and seeks a proof for this implication.
  • Another participant asserts that the statement is not true and challenges others to find a counterexample.
  • A different participant proposes using contraposition as a method to approach the proof, indicating that showing \( x/y = -1 \) implies \( x \cdot y = -1 \) would suffice.
  • Some participants express confusion about the claim and request clarification on why it is considered false.
  • One participant emphasizes the need to find specific values for \( x \) and \( y \) that satisfy \( x/y = -1 \) while ensuring \( x \cdot y \neq -1 \) to prove the original statement false.
  • A counterexample is provided where \( x = -2 \) and \( y = 2 \), demonstrating that \( x \cdot y = -4 \) (not -1) while \( x/y = -1 \). This serves as a challenge to the original claim.
  • Another participant suggests that finding a counterexample might be the easiest approach, proposing that numbers with the same absolute value but opposite signs could serve this purpose.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views on the validity of the original statement, with some asserting it is false and others exploring proof strategies.

Contextual Notes

Participants express uncertainty regarding the implications of the statements and the existence of counterexamples, indicating a need for further exploration of definitions and conditions involved.

wfc
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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.
 
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You don't prove it, because it isn't true. Can you find a couterexample ?
 
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##.
 
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?
 
If you can't come up with an example where that would work, then that means it's false.
 
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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
 
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wfc said:
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
 
FL0R1 said:
if x ≠ -1/y then (-1/y)/y ≠ -1
No, this does not follow at all.
 
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wfc said:
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.
 
  • #10
Svein said:
x⋅y = -4 (which is not -1)
You don't leave any stone unturned :)
 
  • #11
wabbit said:
You don't leave any stone unturned :)
I am a mathematician. I have to turn them.
 
  • #12
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.
 

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