Understanding the Power Map in Tensor Analysis: Bishop and Goldberg Page 6

[pmb_phy]

I'm a bit confused as to how the text Tensor Analysis on Manifolds, by Bishop and Goldberg on page 6.

The authors define the term power set as follows
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If A is a set, we denote by PA the collection of all subsets of A, PA = {C| C is a subset of A}. PA is called the power set of A.
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The authors define the term power map as follows
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If f: A -> B, the we define the power map of f, f: PA -> PB by fC = {fc| fa is an element of C} for every C which is an element of PA}
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What is confusing to me is that nowhere in the definition does the set B occur. What role does B have in the power map?

Thank you

Pete

 

[Doodle Bob]

The authors define the term power map as follows
_________________________________________

If f: A -> B, the we define the power map of f, f: PA -> PB by fC = {fc| fa is an element of C} for every C which is an element of PA}
_________________________________________

What is confusing to me is that nowhere in the definition does the set B occur. What role does B have in the power map?

I think your definition of the power set is a bit off here. Given a mapping f:A->B, where A and B are arbitrary sets, the power map is given by

f(C)={f(c): c is an element of C} for any subset C of A.

The sets A & B are just given sets. The power map is simply the concept that, if you are given a mapping from one set to another, A to B, then this mapping can be used to construct a mapping on the power set of A & B, i.e., one that sends each subset of A to some subset of B.

 

[pmb_phy]

I think your definition of the power set is a bit off here. Given a mapping f:A->B, where A and B are arbitrary sets, the power map is given by

f(C)={f(c): c is an element of C} for any subset C of A.

The sets A & B are just given sets. The power map is simply the concept that, if you are given a mapping from one set to another, A to B, then this mapping can be used to construct a mapping on the power set of A & B, i.e., one that sends each subset of A to some subset of B.

Thanks. But the set B does not appear in the definition of the power map, hence my question.

Pete

 

[Doodle Bob]

Thanks. But the set B does not appear in the definition of the power map, hence my question.

Well, it is there, even if it isn't specifically stated. For any subset C of A and any element c of C, f(c) will be an element of B (since f is a given function from A to B); and hence the image of the subset f(C) will be a subset of B. Hence, the power map is indeed a mapping from the power set of A to the power set of B.