Understanding Index Sets & Union of Sets

[annoymage]

Homework Statement

1.Given a set T we say that T serves as an index set for family F={A_a} of sets if for every a in T there exists a set A_a in family F.

2. By the union of the sets A_a, where a is in T, we mean the set
{x l x$$\in$$A_a for at least one a in T}. We shall denote it by $$\bigcup$$(_a$$\in$$T) A_a.

i think its better if i show example

example:

if S is the set of real number, T is the set of rational number, let, for

a$$\in$$T, A_a = {x$$\in$$S l x$$\geq$$a}

so $$\bigcup$$(_a$$\in$$T) A_a = S

what i don't understand

i can see how ..,A_-1, A₀, A₁ ,... is,

and i don't know how to change it to
$$\bigcup$$(_a$$\in$$T) A_a
or how it is equal to S, because I am perplexed with definition (2) particularly x$$\in$$A_a for at least one a in T

help help

p/s: sorry if it is abit messy, still progressing in latex

 

[HallsofIvy]

Homework Statement

1.Given a set T we say that T serves as an index set for family F={A_a} of sets if for every a in T there exists a set A_a in family F.

Doesn't the definition say "for every a in T there exists exactly one set A_a in family F"?

2. By the union of the sets A_a, where a is in T, we mean the set
{x l x$$\in$$A_a for at least one a in T}. We shall denote it by $$\bigcup$$(_a$$\in$$T) A_a.

i think its better if i show example

example:

if S is the set of real number, T is the set of rational number, let, for

a$$\in$$T, A_a = {x$$\in$$S l x$$\geq$$a}

so $$\bigcup$$(_a$$\in$$T) A_a = S

what i don't understand

i can see how ..,A_-1, A₀, A₁ ,... is,

Do you mean what those sets are?
A_-1 is, by this definition, the set of all real numbers greater than or equal to 0: $\{x| x\ge -1\}= [-1, \infty)$. $A_0= [0, \infty)$, etc.

and i don't know how to change it to
$$\bigcup$$(_a$$\in$$T) A_a

What do you mean "change" it to that? They are not the same at all- one is a collection of sets, the other is the union of all those- the set of all numbers in anyone of them.

or how it is equal to S, because I am perplexed with definition (2) particularly x$$\in$$A_a for at least one a in T

That simply means "x is in at least one of those sets". The union of a collection of sets is, as usual, the set of all members of any of the sets in the collection.

help help

p/s: sorry if it is abit messy, still progressing in latex

In this particular case, since, given any real number x, there exist a rational number, r< x, $x\in A_r$ for that particular r. Since every real number is in at least one of those sets, S is just the set of all real numbers.

 

[weaselman]

It's quite simple really, just a lot of fancy notation.
So, you got a bunch of As, each contains a bunch of x's. Get all the unique x's, from all the A's and put them together into a new set. This new set is S. It contains all the x's such that each x belongs it some of the A's.

I hope it helps.

 

[annoymage]

Doesn't the definition say "for every a in T there exists exactly one set A_a in family F"?

no, i copy this from "Topic in Algebra, Herstien",
but yea, the "there exist a set" confuses me, like, for every a , there's maybe other branches of set.
ok, i'll follow "exactly one".

Do you mean what those sets are?
A_-1 is, by this definition, the set of all real numbers greater than or equal to 0: $\{x| x\ge -1\}= [-1, \infty)$. $A_0= [0, \infty)$, etc.

yes, yes, that's what i mean, I'm still progressing in english, so, correct me if I'm wrong along the way.

What do you mean "change" it to that? They are not the same at all- one is a collection of sets, the other is the union of all those- the set of all numbers in anyone of them.

my mistake

In this particular case, since, given any real number x, there exist a rational number, r< x, $x\in A_r$ for that particular r. Since every real number is in at least one of those sets, S is just the set of all real numbers.

ok, so i confused here,

A(-1)=[-1,oo), A(0)=[0,oo], A(1)=[1,oo),.. these are the sets,

"Since every real number is in at least one of those sets"

so, let 1 for example, A(2) don't have {1}?

correct me please,

now i need to sleep, its 5am here still not sleeping

thanks in advance

 

[annoymage]

That simply means "x is in at least one of those sets". The union of a collection of sets is, as usual, the set of all members of any of the sets in the collection.

now i woke up, i get it already. thank you very much :D