When are two sets the same? When do we say 2 sets are the same?
More specifically, are the following sets same?
$A=\{1,2,3,4,5\}$ 
$B=\{\{1\},\{2,3\},\{4,5\}\}$
Here, $A$ is a set and $B$ is a partition of $A$. 
 A: Two sets are the same if and only if they contain the same elements.
So in this case, those sets cannot be the same. One set has five elements, the other has three. Moreover, one set has numbers as elements, the other has sets as elements.
A: Note : the question that is asked to you aims at two things 


*

*making sure you have understood what  " set identity " means 

*making sure you have understood that : " an element of an element of a set S is not necessarily an element of S". Here, for example, 2 is an element of {2,3}, {2,3} is an element of B; however, the number 2 is NOT an element of B


*The official definition of identity ( i.e. equality) for a set A and a set B is : 

A = B if and only if all the elements of A are also elements of B,
  and reciprocally.



*

*So, the test for equality amounts to asking these 2 questions : 


(1) can I find at least one element of A that is not an element of B? 
(2) can I find at least an element of B that is not an element of A. 


*

*As soon as you can answer "yes" to at least one of these two questions, you can be sure that these 2 sets do not pass the test for equality. 


Note : Of course, you also have to exhibit at least an element that is not a member of both sets to justify your " yes". 


*

*Only if you answer " no" to both questions can you say that the sets A and B actually pass the test and are in fact one and the same set. 


For example, I can find at least an element of set A that is not an element of set B , say the number 2. The number 2 is not an element of B, it is an element of an element of B, namely {2,3}. 
So I am done : A and B are not the same set. 
. 
A: The two sets need to have the same elements. i.e. $$x\in A\Leftrightarrow x\in B.$$
A: Two sets, A and B, are the same, or rather, equal, if each element of A belongs to B and vice versa. In other words, the elements of both the sets should be the same.
If we consider the given example, elements of A are: 1, 2, 3, 4, 5. The elements of B are: {1}, {2,3}, {4, 5}. Clearly, even the number of elements of the two sets are not the same, let alone the elements themselves. Hence the two sets are not the same. 
