union and intersection of family sets

Mar 2012
2
0
Iran
Solved! union and intersection of family sets

Hi there

I'm new here, but I guess I'm going to ask many questions in the future. I'm trying to study mathematics on my own, but I need some help to make sure I'm understanding the concepts correctly...

Anyways, here is my first problem. I'm reading some book and it is mentioned:

The concepts of union and intersection extend readily to large, possibly infinite, collections of sets. Suppose that "I" is some nonempty set (called an index set) and that for each i ∈ I we are given a set Ai ... The union of this family of sets (denoted
) is the set of elements that are in at least one of the sets Ai, that is:

= {x|x ∈ Aj for some j ∈ I}

Where did the "j" come from???

I'm new to this kind of writing algebra and I guess my problem may be that I don't understand the "notation" perfectly. I mean, it's a bit vague. I think it is saying:

The union of this family of sets is equal to all the x members which belong to Aj and this "j" belongs to the set "I".

Now I don't understand; does this mean "the set of elements that are in at least one of the sets Ai"? And if yes, then what's the role of "j"? Where is the "at least" written?

I'll highly appreciate if you give me some general info about the union and intersection of such a family of sets and the way the notation is supposed to be read. I can't establish a connection from the notation to the book's statement in my mind.

This question may not belong here, but when I was searching the forum to find out if someone has asked the same question before, I noticed that almost all questions about sets are asked here.

Thanks a lot
Kindest Regards
Koohyar
 
Last edited:

Plato

MHF Helper
Aug 2006
22,490
8,653
The union of a family of sets is a set to which x belongs if and only if x is in some member of that family.

The intersection of a family of sets is a set to which x belongs if and only if x is in every member of that family.
 
  • Like
Reactions: 1 person

emakarov

MHF Hall of Honor
Oct 2009
5,577
2,017
A mathematical expression may contain variables. In general, the value of such expression can be determined only after all variables are given their concrete values. For example, to compute x + 2y one needs to provide the values of x and y. Similarly, given a sequence \(\displaystyle a_1=5\), \(\displaystyle a_2=4\), \(\displaystyle a_3=3\), \(\displaystyle a_4=2\) and \(\displaystyle a_5=1\), to find the value of \(\displaystyle a_i\) one needs to know the value of \(\displaystyle i\). Variables whose values determine an expression's value are called free.

However, not all variables in an expression are free. Mathematics uses several so-called variable-binding constructs that turn free variables into bound ones, i.e., make variables cease to be free. For example, given the sequence \(\displaystyle a_i\) above, \(\displaystyle \sum_{i=1}^5 a_i=15\); the value of \(\displaystyle i\) is not needed. Bound variables can be renamed without changing the value of an expression; thus, \(\displaystyle \sum_{i=1}^5 a_i=\sum_{j=1}^5 a_j\). In other words, the variable \(\displaystyle i\) is no longer visible from outside of the expression \(\displaystyle \sum_{i=1}^5 a_i\); it only serves to provide internal structure of the expression and is not the expression's input. Examples of other variable-binding constructs are \(\displaystyle \int_0^1 x^2\,dx\), \(\displaystyle \bigcup_{i\in I}A_i\), \(\displaystyle \bigcap_{i\in I}A_i\) and \(\displaystyle \{x\mid x^2+4x-2\le 0\}\).

Two important variable-binding constructs are quantifiers ∀ and ∃. If P is a property, or predicate, (e.g., "is positive"), then ∀x P(x) is true if P(x) holds for all x and is false otherwise. Similarly, ∃x P(x) is true iff P(x) is true for at least some x. Again, given a concrete P, the expressions ∀x P(x) and ∃x P(x) have fixed truth values and don't need x in order to be evaluated.

The set-builder notation {x | P(x)} contains a predicate P that has one free variable x. However, P may also contain bound variables. In the case of \(\displaystyle \{x\mid x\in A_j\text{ for some }j\in I\}\), P(x) is \(\displaystyle \exists j\,(j\in I\text{ and }x\in A_j)\). Thus, P(x) has a concrete truth value for each value of x: P(x) is true if there is some j ∈ I such that \(\displaystyle x\in A_i\). To evaluate \(\displaystyle \bigcup_{i\in I}A_i\), one takes all x for which P(x) is true and collects them into one set.
 
  • Like
Reactions: 1 person
Mar 2012
2
0
Iran
Thanks a lot! (Hi)

I didn't expect such fast and detailed answers in such a short time. You helped me a lot. My eyes are now open on the subject.

Thanks again. (Bow)