Complements of Ranges and Domains

[Kolmogorov]

Given is the function of Set V towards Set W where A is a subset of V and B is a subset of W.

Questions:
Does the range of the complement of A equal the complement of the range of A?
Does the domain of the complement of B equal the complement of the domain of B?I am not entirely sure how to answer this question.

 

[LCKurtz]

Given is the function of Set V towards Set W where A is a subset of V and B is a subset of W.

Questions:
Does the range of the complement of A equal the complement of the range of A?
Does the domain of the complement of B equal the complement of the domain of B?


I am not entirely sure how to answer this question.

Or, apparently, even how to state it. What is B, just any subset of W? If A is a subset what does the "range of A" mean? What is the "domain of a set"?

 

[Kolmogorov]

I am sorry if I didn't formulate the question properly, I had to translate this from Dutch, I don't know if range and domain are the proper terms. The question is about any function in general from V to W without any further specifications.

I would think that these statements are both untrue, because all the elements in Set V and Set W are not necessarily paired, except when we are specifically talking about a bijection. Am I right?

 

[SammyS]

Given is the function of Set V towards Set W where A is a subset of V and B is a subset of W.

Questions:
Does the range of the complement of A equal the complement of the range of A?
Does the domain of the complement of B equal the complement of the domain of B?

I am not entirely sure how to answer this question.

I think that you may mean image rather than range, and pre-image rather than domain.

Giving:

Does the image of the complement of A equal the complement of the image of A?

Does the pre-image of the complement of B equal the complement of the pre-image of B?

 

[Kolmogorov]

Does the image of the complement of A equal the complement of the image of A?

Does the pre-image of the complement of B equal the complement of the pre-image of B?

Yes, that is right.

I didn't know the English translation of these terms, although now I see that the Dutch word is a literal translation of the word image. Pre-image is called the complete image in Dutch.

 

[Kolmogorov]

Searching for preimage I found that the second rule is true: http://mathprelims.wordpress.com/category/topology/page/2/

I think that this must be true, because for every x in V there is only one y. But the other one is not true, because there can be more than one x's in V that have the same y in W.