dmuthuk
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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 X and Y as the set of all ordered pairs (x,y) where x\in X and y\in Y. 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 \{a,b\}, and a function z:\{a,b\}\to X\cup Y such that z(a)\in X and z(b)\in Y. He denotes the set of all such functions from \{a,b\} to X\cup Y as Z. Then, he defines a one-to-one function f:Z\to X\times Y by f(z)=(z(a),z(b)) and claims that the sets Z and X\times Y 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 z\in Z is an ordered pair of the form (x,y)\in X\times Y. Now, since z\in Z itself is a function, we can write z=\{(a,z(a)),(b,z(b))\}=\{\{\{a\},\{a,z(a)\}\},\{\{b\},\{b,z(b)\}\}\} 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 X and Y as the set of all ordered pairs (x,y) where x\in X and y\in Y. 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 \{a,b\}, and a function z:\{a,b\}\to X\cup Y such that z(a)\in X and z(b)\in Y. He denotes the set of all such functions from \{a,b\} to X\cup Y as Z. Then, he defines a one-to-one function f:Z\to X\times Y by f(z)=(z(a),z(b)) and claims that the sets Z and X\times Y 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 z\in Z is an ordered pair of the form (x,y)\in X\times Y. Now, since z\in Z itself is a function, we can write z=\{(a,z(a)),(b,z(b))\}=\{\{\{a\},\{a,z(a)\}\},\{\{b\},\{b,z(b)\}\}\} which doesn't look anything like an ordered pair. So, I am not sure what he means exactly.
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