Can someone explain subsets to me? Not sure how he got this, answers posted

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SUMMARY

The discussion centers on the concept of subsets, specifically regarding the set S = {a, b, c} and its subsets categorized by the number of elements. The subsets are defined as follows: S_0 = {NULL}, S_1 = {{a}, {b}, {c}}, S_2 = {{a, b}, {a, c}, {b, c}}, and S_3 = {{a, b, c}}. The confusion arises from the notation and the understanding of subsets, particularly how the number of elements in each subset corresponds to the index i. The explanation clarifies that S_i represents all subsets of S containing exactly i elements, and the notation requires careful attention to distinguish between sets and their elements.

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Hello everyone I'm confused on how he got this...
the question says:

Let S = {a,b,c} and for each integer i = 0, 1, 2, 3, let S_i be the set of all subests of S that have i elements. List the elements in S_o,S_1,S_2,S_3. Is {S_0,S_1,S_2,S_3} a partion of P(S)?

Here is the answer:

http://suprfile.com/src/1/3rmvhz6/Untitled-1[/URL] copy.jpg[/PLAIN] I know if i = 0,
S_0 = {NULL}
Is it NULL because 0 stands for somthing to be empty?

then I'm really lost on S_1 through S_3
Why did they break S = {a,b,c} into S_1 = {{a},{b},{c}}If i stands for the # of elements then why would S_1 have 3 elements? not 1 element?
S_0, i = 0, means S_0 has 0 elemnets which makes sense to me because NULL means no elements or empty setand for S_2 I'm confused on where they are getting those, its not hte power series because they didn't include {a,b,c} nor did they inlude the NULL set.Then S_3, I'm confused why they just put it all into a separate brackets {{a,b,c}}

Any explanation would be great!I looked up the deinfition of a subset and it really doesn't help me make senes of this, it says a subset is the following:
If A and B are sets, then A is called a subset of B, if and only if, every element of A is also an lelemnt of B.
 
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So we're looking for every set A s.t. if x is an element of A, x is also an element of S.

For S1, you're looking for every set that has one element, such that each element in the set is an element of S (in the subsets, not in S1). So, for example, {a} is a subset of S, because a is an element of S, and {a} also has exactly one member.

On the other hand, if you look at S2, you're looking for subsets of S that have exactly two members. So {a, c} has two members, and a and c are both elements of S, thus it is a two element subset of S.

For S3, the only three element subset of S is clearly {a, b, c}. Here you appear to be confused by the notation... since S3 is a set itself, and {a, b, c} is an element of that set, even though it's also a set itself, you have to put curly braces around the set S3

You should ask your teacher to explain this more carefully to you
 
Thanks Office_Shredder your explanation seems to of made some sense to me.

Our profesor told us this was too easy for him to go over so he skipped lecture on it and continued on, I've sent him a few e-mails but this has helped me a lot more than the e-mails.
 

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