Proving 0 = -0: Axioms & Solutions

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Homework Help Overview

The discussion revolves around proving the statement 0 = -0, which involves exploring axioms related to zero and additive inverses within mathematical structures such as groups or rings.

Discussion Character

  • Conceptual clarification, Assumption checking, Exploratory

Approaches and Questions Raised

  • Participants discuss the axioms that can be used in the proof, questioning the definitions and properties of zero and its additive inverse. There are suggestions to explore the implications of multiplying by -0 and -1, as well as the foundational definitions of -0.

Discussion Status

The discussion is ongoing, with participants raising questions about the appropriate axioms and definitions to use. Some guidance has been offered regarding the properties of zero and the implications of multiplication, but no consensus has been reached yet.

Contextual Notes

Participants are considering the constraints of the mathematical structures involved, such as whether to operate within the real numbers or a more general group context. There is also a focus on the definitions of terms like "-0" and the assumptions that underlie the discussion.

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Homework Statement



prove : 0 = -0


Homework Equations





The Attempt at a Solution


 
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What have you done so far?
 
i do not know how to go about it so that is why i posted it here.
 
If you multiplied both sides of the inequality by -0 and used the distributive property what would you arrive at ? Can you arrive at the same result by using -1 instead ?
 
The crucial question is, what axioms are you allowed to use? The "distributive law" has been suggested but that assumes that you are working in the real numbers or at least a ring in which the distributive law is true. But "0= -0" only requires the "0" element and additive inverse- you should be able to prove this in any group. What is the definition of "-0"? Is it (-1)(0) or "the additive inverse of the multiplicative identity time the additive identity" or just "the additive inverse of the additive identity"?
 
hint:

-0 = -1 * 0

what do you know as axioms about multiplication by zero? :)
 

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