1.3 Extended Set Operations 1 Definition 1.3.7 o baxsbi m sd A Let {A,:a e A} be an indexed collection of sets. The union of the collec tion {A.:a e A} is the set U{A,:a e A} = {x:xe A, for some a A The intersection of the collection {A,:a e A} is the set {A:a e A} : {x:x e A, for every a e A}. CLET %3D Example 1.3.8 Let A = R* and for each a e A let A, = [0, 1 + a). Then {A.iaE A} [0, 1] and U{A.:a e A} = [0, +o0). %3D Example 1.3.9 Let A = Z+ and for each a e A let B, = (-a, a). Then {Ba:a E A} = (-1,1) and U{B,:a e A} = R. || %3D Example 1.3.10 Let A = Z and for each a e A let D, = [a, a + 1]. Then {Da:a e A} = Ø and U{D:a e A} = R. %3D ie product of sets also extends to indexed collections of sets. However. Process is somewhat complicated and will be developed later in the text. the * snghtly different notation is sometimes used for collections of sets. If ben
1.3 Extended Set Operations 1 Definition 1.3.7 o baxsbi m sd A Let {A,:a e A} be an indexed collection of sets. The union of the collec tion {A.:a e A} is the set U{A,:a e A} = {x:xe A, for some a A The intersection of the collection {A,:a e A} is the set {A:a e A} : {x:x e A, for every a e A}. CLET %3D Example 1.3.8 Let A = R* and for each a e A let A, = [0, 1 + a). Then {A.iaE A} [0, 1] and U{A.:a e A} = [0, +o0). %3D Example 1.3.9 Let A = Z+ and for each a e A let B, = (-a, a). Then {Ba:a E A} = (-1,1) and U{B,:a e A} = R. || %3D Example 1.3.10 Let A = Z and for each a e A let D, = [a, a + 1]. Then {Da:a e A} = Ø and U{D:a e A} = R. %3D ie product of sets also extends to indexed collections of sets. However. Process is somewhat complicated and will be developed later in the text. the * snghtly different notation is sometimes used for collections of sets. If ben
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Question
Let {A_alpha: alpha is in Lambda} be an indexed collection of sets.
My question is why we can say that for each beta that is in Lambda, if x is an element of A_beta, then x is an element of bigcup {A_alpha : alpha is in Lambda}. The "Big cup" means the union of the collection {A_alpha: alpha is in Lambda}.
I know I have to use the attached definition (1.3.7), but I would like to know the detail reason.
![1.3 Extended Set Operations
1
Definition 1.3.7 o baxsbi m sd A
Let {A,:a e A} be an indexed collection of sets. The union of the collec
tion {A.:a e A} is the set U{A,:a e A} = {x:xe A, for some a A
The intersection of the collection {A,:a e A} is the set {A:a e A} :
{x:x e A, for every a e A}.
CLET
%3D
Example 1.3.8
Let A = R* and for each a e A let A, = [0, 1 + a). Then {A.iaE A}
[0, 1] and U{A.:a e A} = [0, +o0).
%3D
Example 1.3.9
Let A = Z+ and for each a e A let B, = (-a, a). Then {Ba:a E A} =
(-1,1) and U{B,:a e A} = R.
||
%3D
Example 1.3.10
Let A = Z and for each a e A let D, = [a, a + 1]. Then {Da:a e A} =
Ø and U{D:a e A} = R.
%3D
ie product of sets also extends to indexed collections of sets. However.
Process is somewhat complicated and will be developed later in the text.
the
* snghtly different notation is sometimes used for collections of sets. If
ben](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F44e1b06a-2947-4081-9996-fd6c9a14f5d0%2Fc1346743-8b90-438b-9f01-f2a63e34647a%2Fwbseiah.jpeg&w=3840&q=75)
Transcribed Image Text:1.3 Extended Set Operations
1
Definition 1.3.7 o baxsbi m sd A
Let {A,:a e A} be an indexed collection of sets. The union of the collec
tion {A.:a e A} is the set U{A,:a e A} = {x:xe A, for some a A
The intersection of the collection {A,:a e A} is the set {A:a e A} :
{x:x e A, for every a e A}.
CLET
%3D
Example 1.3.8
Let A = R* and for each a e A let A, = [0, 1 + a). Then {A.iaE A}
[0, 1] and U{A.:a e A} = [0, +o0).
%3D
Example 1.3.9
Let A = Z+ and for each a e A let B, = (-a, a). Then {Ba:a E A} =
(-1,1) and U{B,:a e A} = R.
||
%3D
Example 1.3.10
Let A = Z and for each a e A let D, = [a, a + 1]. Then {Da:a e A} =
Ø and U{D:a e A} = R.
%3D
ie product of sets also extends to indexed collections of sets. However.
Process is somewhat complicated and will be developed later in the text.
the
* snghtly different notation is sometimes used for collections of sets. If
ben
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