MHB Indexed Families of Sets .... Just and Weese .... Exercise 8 ....

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The discussion centers on Exercise 8 from "Discovering Modern Set Theory" by Just and Weese, focusing on Cartesian products and indexed families of sets. The user presents their solution involving the sets A_φ and A_{ { φ } }, and attempts to establish the elements of the Cartesian product A_φ × A_{ { φ } } and the product of indexed sets. They seek confirmation of their calculations and clarity on demonstrating a one-to-one mapping from the product of indexed sets to the Cartesian product. The conversation highlights the need for guidance on the correctness of their approach and the mapping process. Assistance is requested to resolve these conceptual challenges.
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I am reading the book: "Discovering Modern Set Theory. I The Basics" (AMS) by Winfried Just and Martin Weese.

I am currently focused on Chapter 1 Pairs, Relations and Functions ... and I am in particular focused on Cartesian Products and indexed families of sets ...

I need some help with Exercise 8 and some remarks following the exercise ...

The relevant section from J&W is as follows:View attachment 7533
View attachment 7534

It is also worth noting that earlier (on page 12) J&W defined ordered pairs and Cartesian Products as follows:

https://www.physicsforums.com/attachments/7535

I worked Exercise 8 as follows:

Elements of the set $$A_\phi \times A_{ \{ \phi \} }$$


Now ... $$\{ A_\phi = \{ \phi \}$$ and $$ A_{ \{ \phi \} } = \{ \phi , \{ \phi \} \}$$ ...So $$ A_\phi \times A_{ \{ \phi \} } = \{ \langle a,b \rangle \ \mid \ a \in A_\phi \text{ and } b \in A_{ \{ \phi \} } \}
$$

$$= \{ \ \langle \phi , \phi \rangle \ , \langle \phi , \{ \phi \} \rangle \ \} $$

$$ = \{ \ \{ \phi , \{ \phi \} \} \ , \ \{ \phi , \{ \{ \phi \} \} \ \} $$

Elements of the set $$\prod_{ i = \{ \phi , \{ \phi \} } A_i $$


Let $$I = \{ \phi , \{ \phi \} \} $$
$$\prod_{ i \in I } A_i = \{ f \in ( \bigcup \{ A_i \ : \ i \in I \} )^I \ : \ \forall i \in I , \ ( f(i) \in A_i ) \} $$

where $$( \bigcup \{ A_i \ : \ i \in I \} )^I = \{ f \ : \ I \rightarrow \bigcup A_i \} $$

$$= \{ f \ : \ \{ \phi , \{ \phi \} \} \rightarrow \bigcup A_i \} $$
Now ... we have to consider the function(s) from the domain $$\{ \phi , \{ \phi \} \}$$ to

$$ \bigcup A_i = A_\phi \cup A_{ \{ \phi \} } $$

$$= \{ \phi , \{ \phi \} \}$$The only function satisfying the required conditions is the following function f:

$$f = \{ \langle \phi , \phi \rangle , \langle \{ \phi \} , \{ \phi \} \rangle \} $$

$$= \{ \{ \phi , \{ \phi \} \} , \{ \{ \phi \} , \{ \{ \phi \} \} \}$$
My questions as follows:

Can someone please either confirm my working as correct or point out the errors ...?

Further, can someone show me simply and explicitly how H is a one to one map from $$\prod_{ i = \{ \phi , \{ \phi \} \} } A_i $$ onto $$A_\phi \times A_{ \{ \phi \} }$$ ... ... ?

Help will be much appreciated ...

Peter
 
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Peter said:
I am reading the book: "Discovering Modern Set Theory. I The Basics" (AMS) by Winfried Just and Martin Weese.

I am currently focused on Chapter 1 Pairs, Relations and Functions ... and I am in particular focused on Cartesian Products and indexed families of sets ...

I need some help with Exercise 8 and some remarks following the exercise ...

The relevant section from J&W is as follows:It is also worth noting that earlier (on page 12) J&W defined ordered pairs and Cartesian Products as follows:
I worked Exercise 8 as follows:

Elements of the set $$A_\phi \times A_{ \{ \phi \} }$$


Now ... $$\{ A_\phi = \{ \phi \}$$ and $$ A_{ \{ \phi \} } = \{ \phi , \{ \phi \} \}$$ ...So $$ A_\phi \times A_{ \{ \phi \} } = \{ \langle a,b \rangle \ \mid \ a \in A_\phi \text{ and } b \in A_{ \{ \phi \} } \}
$$

$$= \{ \ \langle \phi , \phi \rangle \ , \langle \phi , \{ \phi \} \rangle \ \} $$

$$ = \{ \ \{ \phi , \{ \phi \} \} \ , \ \{ \phi , \{ \{ \phi \} \} \ \} $$

Elements of the set $$\prod_{ i = \{ \phi , \{ \phi \} } A_i $$


Let $$I = \{ \phi , \{ \phi \} \} $$
$$\prod_{ i \in I } A_i = \{ f \in ( \bigcup \{ A_i \ : \ i \in I \} )^I \ : \ \forall i \in I , \ ( f(i) \in A_i ) \} $$

where $$( \bigcup \{ A_i \ : \ i \in I \} )^I = \{ f \ : \ I \rightarrow \bigcup A_i \} $$

$$= \{ f \ : \ \{ \phi , \{ \phi \} \} \rightarrow \bigcup A_i \} $$
Now ... we have to consider the function(s) from the domain $$\{ \phi , \{ \phi \} \}$$ to

$$ \bigcup A_i = A_\phi \cup A_{ \{ \phi \} } $$

$$= \{ \phi , \{ \phi \} \}$$The only function satisfying the required conditions is the following function f:

$$f = \{ \langle \phi , \phi \rangle , \langle \{ \phi \} , \{ \phi \} \rangle \} $$

$$= \{ \{ \phi , \{ \phi \} \} , \{ \{ \phi \} , \{ \{ \phi \} \} \}$$
My questions as follows:

Can someone please either confirm my working as correct or point out the errors ...?

Further, can someone show me simply and explicitly how H is a one to one map from $$\prod_{ i = \{ \phi , \{ \phi \} \} } A_i $$ onto $$A_\phi \times A_{ \{ \phi \} }$$ ... ... ?

Help will be much appreciated ...

Peter
I have been reflecting on my solution to Just and Weese Exercise 8 ...

I know think I may have made an error in determining the set $$\prod_{ i = \{ \phi , \{ \phi \} } A_i $$

... ... so I am now attempting to give a correct solution ...
We have that ... $$ A_\phi = \{ \phi \}$$ and $$ A_{ \{ \phi \} } = \{ \phi , \{ \phi \} \}$$ ...and we let $$I = \{ \phi , \{ \phi \} \} $$
Then ... ... $${}^I \!\left( \bigcup \{ A_i \}\right) = \{ f \ : \ I \rightarrow \bigcup A_i \}$$ But ... ... $$\bigcup A_i = \bigcup \{ A_\phi, A_{ \{ \phi \} } \} = A_\phi \cup A_{ \{ \phi \} } = \{ \phi , \{ \phi \} \}
$$Therefore ... $${}^I \!\left( \bigcup \{ A_i \}\right) = \{ f \ : \ \{ \phi , \{ \phi \} \} \rightarrow \{ \phi , \{ \phi \} \}$$ ... ... (1)and $$\prod_{ i \in I } A_i = \{ f \in {}^I \!\left( \bigcup \{ A_i \}\right) \ : \ \forall i \in I \ (f(i) \in A_i \ ) \ \} $$ ... ... ... (2)So ... working from (1) and (2) we have that $$\prod_{ i \in I } A_i = \{ g, h \}$$where (treating functions as sets of ordered pairs ...)$$g = \{ \ \langle \phi , \phi \rangle \ , \ \langle \{ \phi \} , \phi \rangle \ , \} = \{ \ \{ \phi , \{ \phi \} \} \ , \{ \{ \phi \} , \{ \phi \} \} \}$$and $$h = = \{ \ \langle \phi , \phi \rangle \ , \ \langle \{ \phi \} , \{ \phi \} \rangle \ \} = \{ \ \{ \phi , \{ \phi \} \} \ , \{ \{ \phi \} , \{ \{ \phi \} \} \ \} $$... ... BUT ... ... where to from here ... hmmm ...really need some help/guidance ...In particular how exactly and explicitly do we demonstrate the one-to-one map H from $$\prod_{ i = \{ \phi , \{ \phi \} } A_i $$ to $$A_\phi \times A_{ \{ \phi \} }$$ ... .. Help will be much appreciated ...

Peter
 
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