Dgdrxrt

What is the difference between A=1,2,3,4 and B=1,2,3,4?
why C=1,2,3,1 exists? what does these sets mean ?

In C, 1 is repeated...

Thank you so much

In a nutshell: "the number x" and "the set with $x$ as an element", written $leftxright$, are not the same.

what is the difference between A=1,2,3,4 and B=1,2,3,4?
why C=1,2,3,1 exists? what does these sets mean ?

• 
  $B$ is a set with four elements, those elements are the numbers $1$, $2$, $3$ and $4$;


• 
  $A$ is a set with four elements: those elements are the three numbers $2$, $3$ and $4$ and the set containing $1$ (written $left1right$);


• 
  $C=leftcolorblueleft1,left2rightright,colorgreen3,colorpurpleleft1rightright$ is a set with three elements:
  
  
  • the number $3$ (in green);
  
  
  • the set containing $1$ (in purple);
  
  
  • the set containing: the number $1$ and the set $left2right$ (in blue).
  
  

A contains one set; B contains none.

C contains two sets, and one of these sets contains a set.

A lot of the answers here just answer the question in a way that I'm sure the OP has seen or read before. So I would like to explain in a way that would explain sets to say a 10-year-old. Say you have three cases, the first where you have $2$ bags, the second case where you have $1$ bag and the third case where you have $4$. Now in the first case say you put all of your objects into one bag and then you decide that some of the objects in the bag have common properties and you can group them together, but they still belong to the larger bag, so you take your second bag, put all the common items in and then put that inside the bigger bag.

Now take your second case where you just dump all the objects into one bag and don't bother with any inter-grouping. This is the second set in your question.

Third, you take the $4$ bags and put all the items in one giant bag and then take one item out that does not share any properties with the other but you still feel is important enough to have its own category. Then you find that you can have another one where multiple items can be grouped together and you put these in another bag, inside which you see one thing is important enough once again to have its own place but still belong to this larger group. Now you place the two bags, one with only one special item inside and the other with some items and one special item kept inside that bag in its own bag.

I know, got a little confusing there but if you understood what I was trying to do then you'll know what I'm talking about, This is analogous to sets and just common elements. The common elements are like all things just placed inside a big bag and the ones that have set brackets around them within the sets are the ones that are placed inside smaller bags and put inside the larger bag along with the rest of the items.