Are {a,b,c} and {a,{b,c}} the same set?

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Discussion Overview

The discussion centers around the question of whether the sets {a,b,c} and {a,{b,c}} are equal. Participants explore the definitions and properties of sets, particularly focusing on the elements contained within each set and the implications of set size.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants assert that the two sets are not equal due to differing elements, with {a,b,c} containing three distinct elements and {a,{b,c}} containing two elements.
  • It is noted that for sets to be equal, they must have exactly the same elements, which is not the case here.
  • One participant points out that while {b,c} is a member of {a,{b,c}}, it is also a subset of {a,b,c}, highlighting a potential confusion between membership and subset relations.
  • Another participant questions whether there is an equivalence relation for sets, suggesting a deeper exploration of set theory concepts.

Areas of Agreement / Disagreement

Participants generally agree that the two sets are not equal due to their differing elements and sizes. However, there is some exploration of related concepts, such as membership and subset relations, which may indicate areas of confusion.

Contextual Notes

The discussion does not resolve the broader implications of set theory or equivalence relations, and it remains focused on the specific comparison of the two sets in question.

tgt
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Does {a,b,c}={a,{b,c}}?
 
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No...
 


Pere Callahan said:
No...

why not?
 


Is there some sort of equivalence relation for sets?
 


To sets are equal if they have the same elements. The set on the left hand side of your equation has elements, a,b,c, the one on the right hand side has elements a, {b,c} which are clearly differnent.
 


tgt said:
Does {a,b,c}={a,{b,c}}?

{a, b, c} contains three elements.

{a, {b, c}} contains two elements.

Sets of different sizes are never equal.

{b, c}, the second element listed in the second set, it a set, but it still counts as a single element.
 


Or, to say the same thing in a slightly different way:

{a, b, c} is a set that has three elements: a, b, and c.

{a, {b,c}} is a set that has two elements: a and {b,c}.

In order for sets to be equal, they must have exactly the same elements.

Notice, also that while {b,c} is a member of {a, {b, c}}, it is a subset of {a, b, c}.

[tex]\{b,c\}\in \{a,\{b,c\}\}[/tex]
[tex]\{b,c\}\subset \{a, b, c\}[/tex]

You may confusing those two concepts.-
 

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