- #1
dmuthuk
- 41
- 1
Source: Halmos, Naive Set Theory
I ran into a bit of confusion in the way Halmos generalizes the "Cartesian Product" for a family of sets (p.36). I was wondering if someone can shed some light on this. Here is my problem:
Previously, Halmos defines the cartesian product of two sets [tex]X[/tex] and [tex]Y[/tex] as the set of all ordered pairs [tex](x,y)[/tex] where [tex]x\in X[/tex] and [tex]y\in Y[/tex]. On p.36, in order to generalize the concept, he introduces a new way of looking at cartesian products.
I will stick to our special case of two sets for comparison. First, he considers an arbitrary unordered pair of sets [tex]\{a,b\}[/tex], and a function [tex]z:\{a,b\}\to X\cup Y[/tex] such that [tex]z(a)\in X[/tex] and [tex]z(b)\in Y[/tex]. He denotes the set of all such functions from [tex]\{a,b\}[/tex] to [tex]X\cup Y[/tex] as [tex]Z[/tex]. Then, he defines a one-to-one function [tex]f:Z\to X\times Y[/tex] by [tex]f(z)=(z(a),z(b))[/tex] and claims that the sets [tex]Z[/tex] and [tex]X\times Y[/tex] are essentially the same and only differ in notation. However, I don't know why this is the case.
If this is true, then each [tex]z\in Z[/tex] is an ordered pair of the form [tex](x,y)\in X\times Y[/tex]. Now, since [tex]z\in Z[/tex] itself is a function, we can write [tex]z=\{(a,z(a)),(b,z(b))\}=\{\{\{a\},\{a,z(a)\}\},\{\{b\},\{b,z(b)\}\}\}[/tex] which doesn't look anything like an ordered pair. So, I am not sure what he means exactly.
I ran into a bit of confusion in the way Halmos generalizes the "Cartesian Product" for a family of sets (p.36). I was wondering if someone can shed some light on this. Here is my problem:
Previously, Halmos defines the cartesian product of two sets [tex]X[/tex] and [tex]Y[/tex] as the set of all ordered pairs [tex](x,y)[/tex] where [tex]x\in X[/tex] and [tex]y\in Y[/tex]. On p.36, in order to generalize the concept, he introduces a new way of looking at cartesian products.
I will stick to our special case of two sets for comparison. First, he considers an arbitrary unordered pair of sets [tex]\{a,b\}[/tex], and a function [tex]z:\{a,b\}\to X\cup Y[/tex] such that [tex]z(a)\in X[/tex] and [tex]z(b)\in Y[/tex]. He denotes the set of all such functions from [tex]\{a,b\}[/tex] to [tex]X\cup Y[/tex] as [tex]Z[/tex]. Then, he defines a one-to-one function [tex]f:Z\to X\times Y[/tex] by [tex]f(z)=(z(a),z(b))[/tex] and claims that the sets [tex]Z[/tex] and [tex]X\times Y[/tex] are essentially the same and only differ in notation. However, I don't know why this is the case.
If this is true, then each [tex]z\in Z[/tex] is an ordered pair of the form [tex](x,y)\in X\times Y[/tex]. Now, since [tex]z\in Z[/tex] itself is a function, we can write [tex]z=\{(a,z(a)),(b,z(b))\}=\{\{\{a\},\{a,z(a)\}\},\{\{b\},\{b,z(b)\}\}\}[/tex] which doesn't look anything like an ordered pair. So, I am not sure what he means exactly.
Last edited: