Indexed family? smart enuff 2b dumb enuff?

[SqrachMasda]

this seems extremely confusing to me
so if anybody understands it they're a genius to me
if anybody can dumb it down and actually explain it to me, they're GOD

it was briefly introduced in my text on about half a page and was completely confusing
the only practice problem ended in an empty set {0} which made it no help
my professor breezed through it at the end of class but was also compared it with unions and intersections which the symbols look exactly the same when humans draw it

it was way too much with way to little information
i have always had a big problem grasping set theory and proofs with such
I think because there is so many symbols I have to remember but usually just get mixed up

so, if anybody can explain this in dumb terms and then maybe relate it again in the correct terms
i will place them in a whole other level of intelligence
even if it's not important, i have to understand why i can't understand it
the only thing i picked up on from class was instead of unions and intersections with 2 or 3 sets (which i pretty much understand) it is based on that with a lot or even infinite sets
but then why so many new symbols

i'm tearing my hair out because the limited information i have is no help

i been here before so i already expect the arrogant smart people will grunt at me on this :grumpy:

 

[EnumaElish]

If you know arithmetic, you can understand this.

"1 add 2" is written "1 + 2". Suppose you wish to add all the numbers from 1 to an unspecified integer N. You'd write that as 1 + ... + N. As a shorthand, you might say "sum of all k's from k=1 to k=N." Here, k denotes each number from 1 through N, taken consecutively. The standard notation for this is $\sum_{k=1}^N k$. Think of the symbol $\sum$ as a large "plus" symbol. (Actually $\sum$ is the Greek letter Sigma corresponding to the Latin letter S, which is the first letter of the English word "Sum.")

Similarly for two sets S₁ and S₂, "S₁ union S₂" is written "S₁ U S₂." Suppose you wish to unionize all the sets from S₁ to S_N, where N is an unspecified integer (and each such set is being defined elsewhere). You'd write that as S₁ U ... U S_N. As a shorthand, you might say "union of all S_k from k=1 to k=N." Just like above, k denotes each number (index) from 1 through N, taken consecutively. The standard notation for this is $\cup_{k=1}^N S_k$.

Example 1: Let S_k={k}. Then $\cup_{k=1}^N S_k$ = {1, ..., N}.
Example 2: Let S_k={100}. Then $\cup_{k=1}^N S_k$ = {100}.

 

[SqrachMasda]

okay, i see how it is similar to summations...
gotta run to work
but i want to try and work out a problem from my text later tonite
if not maybe put it up here
because the notation they use is absurd to me

 

[verty]

For me, the most intuitive way to understand these is from an inductive (recursive?) definition. Please excuse my notation:

Sigma(n=a->a)(f(x)) = f(a)
Sigma(n=a->b+1)(f(x)) = Sigma(n=a->b)(f(x)) + f(b+1)

Union(k=1->1)(S_k) = S_1
Union(k=1->n+1)(S_k) = Union(k=1->n)(S_k) U S_(n+1)

 

[matt grime]

That only works if the index set is well ordered, and as written, actually a finite set.