Symmetric Difference if/then Proof

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Discussion Overview

The discussion revolves around proving the statement "if X ⊕ Y = Y ⊕ X then X = Y" within the context of symmetric difference operations on sets. Participants explore the properties of symmetric difference, including commutativity and associativity, and seek clarification on the definitions involved.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant seeks guidance on proving the statement involving symmetric difference and requests an explanation suitable for someone with limited proof skills.
  • Another participant questions the meaning of "oplus" and the nature of the sets X and Y, indicating a need for clarification on the terms used.
  • A later reply clarifies that X and Y are sets and that "oplus" refers to the symmetric difference operation, which is defined as the XOR operation.
  • One participant expresses skepticism about the validity of the statement being proven, noting that both symmetric difference and exclusive or operations are commutative, implying that the equality holds regardless of the equality of X and Y.

Areas of Agreement / Disagreement

Participants do not reach a consensus. There is disagreement regarding the truth of the statement being proven, with some asserting that the commutative property undermines the claim.

Contextual Notes

Participants highlight the need for clarity on definitions and properties of the operations involved, as well as the specific nature of the sets in question. The discussion reflects uncertainty about the implications of the properties of symmetric difference.

pingi
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Hi there,

I'm trying to figure out proving the following:
if X oplus Y = Y oplus X then X = Y

In order to prove it, I need to use the symmetric difference associativity & other characteristics and identities.

Can you please give me a direction?
Please explain the answer as a teacher would, as my skills of proving this kind of arguments are poor.

Thanks! Pingi.
 
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What is oplus? Also what kind of things are X and Y (sets, logical variables, other things)?
 
Hi mathman,

Sorry for being unclear about my question and thanks for directing me!

1. X and Y are sets.
2. 'oplus' is an add symbol in circle - ⊕, used to describe the symmetric difference of the two sets (with the XOR operation).

Pingi.
 
If it is referring to the symmetric difference or to the exclusive or operations, then I actually don't believe the statement you are trying to prove is true. Both of these operations are commutative, which means that:

X "oplus" Y = Y "oplus" X

regardless of what X and Y are.

If "oplus" means something different, however, then please explain it in more depth.
 

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