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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
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
Last edited: