Confused on directions, List the elements in the subsets?

[mr_coffee]

Hello everyone. There arn't any problems like this in this section, so I'm kind of lost on what they want. It says...

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

S_0 = {a_0,b_0,c_0}
S_1 = {a_1,b_1,c_1}

or what are they saying exactly?

Isn't S_0 through 3 going to have the exact same elements just {a,b,c} all the time? I don't see how changing the subscript on S is changing the elements or the number of elements. I have no idea what they want, The only thing i saw in the book was the folllowing:
THe number of subsets of a set.

THe following theorem states the important fact that if a set has n elements, then its power set has 2^n elements.
Suppose X is a set and z is an elemen tof X.If X = {x,y,z}, the following table shows the correspondence between subsets of X that do not contain z and subsets of X that contain Z.

The table shows subset so X that do not contain z (X- {z} )
Null
{x}
{y}
{x,y}

Subsets of X that contain z
NULL union {z} = {z}
{x} union {z} = {x,z}
{y} union {z} = {y,z}
{x,y} union {z} = {x,y,z}

But i don't see how this relates to my probem at all.

Any help would be great

 

[radou]

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

S_0 = {a_0,b_0,c_0}
S_1 = {a_1,b_1,c_1}

or what are they saying exactly?

For example, for i = 2, S2 = {{a,b}, {b, c}, {a, c}}.

http://mathworld.wolfram.com/PowerSet.html"

 

[mr_coffee]

Thanks for the responce but that's what confusees me, i know what a power set is, like if you find
P({x,y}) = {NULL, {x},{y}, {x,y}}
but

Are you taking the power set of S? in ur example?
or how did u get {{a,b},{b,c},{a,c}} if that was a power set, isn't it suppose to be 2^n, so if i = 2, wouldn't u have 4 sets in there?

When they say list the elments in S0 - S3, do they mean take the power set of S0 - S3? and if that's the case, how does the subscript affect what's in S = {a,b,c}?