Hello everyone I am new to the site, I have a question that was in the test and did not understand the parts that are in the question. This question from a test I failed to pass, in a machine learning course. the question:

Suppose that $$F\subseteq\{0,1\}^\Omega$$ is some collection of Boolean functions over $$\Omega$$.

Define $$\Omega'=\Omega\times\{0,1\}$$ and define $$F'$$ to be the collection of Boolean functions over $$\Omega'$$ as follows:

for every $$f\in F$$, there is some $$f'\in F'$$ that maps $$(x,y)\in\Omega'$$ to $$1[f(x)\neq y]$$.
(Furthermore, $$F'$$ consists only of such $$f'$$; no other functions are allowed.)

Prove that the VC-dimensions of $$F$$ and $$F'$$ are equal.

I think this is an easy question, but there are concepts and things I did not understand in the question itself. I tried to search for information on the internet but was unsuccessful

I did not understand it: what does this marking mean? $$\Omega'=\Omega\times\{0,1\}$$

I did not understand what x and y are, what do they mean? And what is the function with the number 1, what does this function do? $$(x,y)\in\Omega'$$ to $$1[f(x)\neq y]$$

Another question What is the VC dimension? Is it like a two-dimensional plane or a 3-dimensional space?

The question itself I think I will be able to solve only when I understand the question itself, I tried to understand the question, but I could not.

• Are you familiar with the definition of VC dimension? If not, you won't be able to solve questions about VC dimension. Oct 23 '21 at 21:41
• The notation $\Omega \times \{0,1\}$ stands for the Cartesian product of sets. Once you understand what this means, you will know what $x,y$ are. Oct 23 '21 at 21:41
• The notation $1[B]$ stands for $1$ if $B$ holds and $0$ if $B$ doesn't hold. Oct 23 '21 at 21:42
• cs.stackexchange.com/q/144960/755
– D.W.
Oct 26 '21 at 3:35
• Please try to ask questions that will be useful to others, even if they are not looking at exactly the same exercise/question that you are.
– D.W.
Oct 26 '21 at 3:36