What is an indexed family of sets. I need a simple example

[Ahmed Abdullah]

I have looked it in the Wikipedia, but no simple example. So I am not sure. Is the indexed family of sets just power sets, indexed (indexing means labeling as I understand)?

For example the indexed family of sets of set A ={1,2,3,4,5,6} is just the collection of element from power set. A sub 1 may be {1} and A sub 7 may be {1,2} and so on. Indexed family of sets may be the collection of those sets as I understand. Can anyone clarify this please.(I am not a math major.)

 

[LCKurtz]

Here's an example that may help. For r > 0 define$$A_r = \{(x,y):x^2 + y^2 < r^2\}$$This gives an uncountable family of nested discs indexed by their radius.

 

[ArcanaNoir]

Let the group of sets be called G. In G, there are five sets, G1, G2, G3, G4, and G5. Let those sets be the following:
G1: {2, 4, 6, 8,}
G2: {3, 6, 9, 12}
G3: {4, 8, 12, 16}
G4: {5, 10, 15, 20}
G5: {6, 12, 18, 24}

So, G: {G1, G2, G3, G4, G5}

This is a family of sets. I think the index refers to the sub number. In paper/pencil land for G1, the 1 would be a subscript.

 

[HallsofIvy]

On this board, you an do it with the html code C[ sub]1[ /sub ] without the spaces: C₁. Or do it using the tex code: [ tex ]C_1[ /tex ] without the spaces gives $C_1$.

 

[Joffan]

{Alice, Bob, Carla} share a house, but they're not always all in. The set of possible occupant sets of the house can be indexed [0..7] with A+2B+4C.

 

[Ahmed Abdullah]

Let the group of sets be called G. In G, there are five sets, G1, G2, G3, G4, and G5. Let those sets be the following:
G1: {2, 4, 6, 8,}
G2: {3, 6, 9, 12}
G3: {4, 8, 12, 16}
G4: {5, 10, 15, 20}
G5: {6, 12, 18, 24}

So, G: {G1, G2, G3, G4, G5}

This is a family of sets. I think the index refers to the sub number. In paper/pencil land for G1, the 1 would be a subscript.

Is it just the family of set with index notation? Wiki gave me "Let S be a set. An indexed family of sets {C_i}_iεI is an indexed family that maps I to elements of the power set of S.

Hence, an indexed family of sets is conceptually different from a family of sets (which is just a synonym for "set of sets"), but in practice the distinction is sometimes fuzzy and the indexed family is identified with its range and treated like an ordinary family."

 

[ArcanaNoir]

Ah, I messed up. I will now use my textbook to define it.

"Let $\Delta$ be a non-empty set such that for each $\alpha$ $\in$ $\Delta$ there is a corresponding set A_$\alpha$. The family {A_$\alpha$: $\alpha$ $\in$ $\Delta$} is an indexed family of sets. The set $\Delta$ is called the indexing set and each $\alpha$ $\in$ $\Delta$ is an index."

So my delta was {1, 2, 3, 4, 5}. That is, my indexing set was {1, 2, 3, 4, 5}. It could have easily been all natural numbers or some other known set, and then I could say my indexing set was N, that is, Natural Numbers. My family is all G_i, such that i is an element of {1, 2, 3, 4, 5}, that is, my family is: {G_i: i $\in$$\Delta$} (if I want my set {1, 2, 3, 4, 5} to be named delta. it doesn't have to be named that.) . Thus, my family is {G₁, G₂, G₃, G₄ G₅}. The numbers 1, 2, 3, 4, and 5 are the indices. Each is an index.

 

[Ahmed Abdullah]

Thanks ArcanaNoir for the response. I also like LCKurtz' example where Δ is all positive real number.