Prove or find counterexamples

  • Context: Undergrad 
  • Thread starter Thread starter mbcsantin
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Discussion Overview

The discussion revolves around the associative property of the binary product involving sets, specifically examining the equality (A x B) x C = A x (B x C). Participants are exploring the proof or potential counterexamples related to this property, with some confusion regarding the context of sets versus real numbers.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses uncertainty about how to begin proving the equality of the binary product.
  • Another participant suggests that to prove the equality, one would typically show that both inclusions are true, questioning if the elements can be represented as ((a,b),c) = (a,(b,c)).
  • A later reply asserts that the elements are indeed equal and references the Associative Property of Multiplication, stating that for real numbers a, b, and c, the grouping does not affect the product.
  • Another participant clarifies that they misunderstood the context, initially thinking the discussion was about the Cartesian product rather than the associative property of multiplication.
  • One participant challenges the previous assertions by pointing out the inconsistency in the context, highlighting the shift from discussing sets to real numbers and questioning the clarity of the original question.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the context of the discussion, with some focusing on sets and others on real numbers. The clarity of the original question remains unresolved.

Contextual Notes

There is a lack of clarity regarding the definitions being used, particularly the distinction between sets and real numbers, which may affect the understanding of the associative property in this context.

mbcsantin
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..using only the definition of the binary product:

for any sets A, B, C in a universe U:

(A x B) x C = A x (B x C)

I have no clue how to even get started with this one. Somebody help me please!
 
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Typically you would want to show that both inclusions are true...but in this case...is the element [tex]((a,b),c) = (a,(b,c))[/tex] ?
 
daveyinaz said:
Typically you would want to show that both inclusions are true...but in this case...is the element [tex]((a,b),c) = (a,(b,c))[/tex] ?

Yes, they're equal. It's called the Associative Property of Multiplication.

The property which states that for all real numbers a, b, and c, their product is always the same, regardless of their grouping:
(a . b) . c = a . (b . c)
 
I'm sorry..I didn't read the your post correctly if that's the case..was thinking cartesian product.
 
mbcsantin said:
Yes, they're equal. It's called the Associative Property of Multiplication.

The property which states that for all real numbers a, b, and c, their product is always the same, regardless of their grouping:
(a . b) . c = a . (b . c)

Then please go back and ask whatever question you are REALLY asking. In your original post, A, B, and C are sets. Now you are telling us that they are real numbers. Also in your first post you asked about proving "A x(B x C)= (A x B)x C" but now you are saying that is the "Associative Property of Multiplication" which apparently you are accepting as true. At this point, I have no idea what your question really is!
 

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