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

  • Thread starter tgt
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In summary: Sets are equal if they have the same elements and are of the same size. Therefore, {a,b,c} and {a,{b,c}} are not equal because they have different sizes and elements, with {a,{b,c}} containing a set as one of its elements. This difference in size and elements makes them two distinct sets and they cannot be equal.
  • #1
tgt
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Does {a,b,c}={a,{b,c}}?
 
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  • #2


No...
 
  • #3


Pere Callahan said:
No...

why not?
 
  • #4


Is there some sort of equivalence relation for sets?
 
  • #5


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.
 
  • #6


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.
 
  • #7


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.-
 

1. What is the meaning of {a,b,c} and {a,{b,c}}?

The notation {a,b,c} represents a set with the elements a, b, and c. The notation {a,{b,c}} represents a set with two elements, a and a set containing b and c.

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

No, they are not the same set. The first set has three elements, while the second set has two elements. The second set also contains a set as one of its elements, while the first set does not.

3. Can a set contain another set as one of its elements?

Yes, a set can contain another set as one of its elements. This is known as a nested set. In the example {a,{b,c}}, the set {b,c} is nested within the larger set.

4. Why is it important to understand the difference between {a,b,c} and {a,{b,c}}?

Understanding the difference between these two notations is important in set theory and mathematics in general. It helps to avoid confusion and ensures accuracy when working with sets and their elements.

5. Can the elements of a set be in a specific order?

No, the elements of a set are not in a specific order. Sets are considered unordered collections of elements, so the order in which the elements are listed does not matter. In the example {a,b,c} and {c,b,a} are the same set.

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