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I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ...

I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.2 Relations and Functions ...

I need some help in fully understanding some remarks by Searcoid in his sub-section on products ...

The subsection on products reads as follows:View attachment 7513

https://www.physicsforums.com/attachments/7514

In the above text by Searcoid we read the following:

" ... ... Suppose \(\displaystyle (X_i)_{ i \in I }\) is a family of sets . Then \(\displaystyle \bigcup (X_i)\) is a set by Axiom IV, whence \(\displaystyle ( \bigcup (X_i) )^I\) is a set by (1.2.11) and \(\displaystyle \{ x \in ( \bigcup (X_i) )^I \ \lvert \ \forall j \in I , x_j \in X_j \}\) ... ... "I do not understand the form and nature of \(\displaystyle ( \bigcup (X_i) )^I\) ... ... what is meant by this? Can someone please give a simple and clear explanation ... and perhaps an example or two ...

Further, likewise I do not understand the form and nature of \(\displaystyle \{ x \in ( \bigcup (X_i) )^I \ \lvert \ \forall j \in I , x_j \in X_j \}\) ... ... again, what is meant by this? Can someone please give a simple and clear explanation ... and perhaps an example or two ... Help will be much appreciated ... ...

Peter=========================================================================================Section 1.2.11 is mentioned in the above quote ... so I am proving the text of the same ... as follows:https://www.physicsforums.com/attachments/7515

View attachment 7516Searcoid's sub-section on families of sets may also be relevant so I am providing the same ... as follows:https://www.physicsforums.com/attachments/7517

https://www.physicsforums.com/attachments/7518

Hope someone can help ...

Peter

I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.2 Relations and Functions ...

I need some help in fully understanding some remarks by Searcoid in his sub-section on products ...

The subsection on products reads as follows:View attachment 7513

https://www.physicsforums.com/attachments/7514

In the above text by Searcoid we read the following:

" ... ... Suppose \(\displaystyle (X_i)_{ i \in I }\) is a family of sets . Then \(\displaystyle \bigcup (X_i)\) is a set by Axiom IV, whence \(\displaystyle ( \bigcup (X_i) )^I\) is a set by (1.2.11) and \(\displaystyle \{ x \in ( \bigcup (X_i) )^I \ \lvert \ \forall j \in I , x_j \in X_j \}\) ... ... "I do not understand the form and nature of \(\displaystyle ( \bigcup (X_i) )^I\) ... ... what is meant by this? Can someone please give a simple and clear explanation ... and perhaps an example or two ...

Further, likewise I do not understand the form and nature of \(\displaystyle \{ x \in ( \bigcup (X_i) )^I \ \lvert \ \forall j \in I , x_j \in X_j \}\) ... ... again, what is meant by this? Can someone please give a simple and clear explanation ... and perhaps an example or two ... Help will be much appreciated ... ...

Peter=========================================================================================Section 1.2.11 is mentioned in the above quote ... so I am proving the text of the same ... as follows:https://www.physicsforums.com/attachments/7515

View attachment 7516Searcoid's sub-section on families of sets may also be relevant so I am providing the same ... as follows:https://www.physicsforums.com/attachments/7517

https://www.physicsforums.com/attachments/7518

Hope someone can help ...

Peter

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