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

[Math Amateur]

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

 

[Math Amateur]

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