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

[tgt]

Does {a,b,c}={a,{b,c}}?

 

[Pere Callahan]

No...

 

[tgt]

No...

why not?

 

[tgt]

Is there some sort of equivalence relation for sets?

 

[Pere Callahan]

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.

 

[Tac-Tics]

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.

 

[HallsofIvy]

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}.

$$\{b,c\}\in \{a,\{b,c\}\}$$
$$\{b,c\}\subset \{a, b, c\}$$

You may confusing those two concepts.-