 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 2

For every fixed $x_{1:n}$,
$$
(\mathcal H_1\cup\mathcal H_2)(x_{1:n})
=\mathcal H_1(x_{1:n})\cup\mathcal H_2(x_{1:n}).
$$
The cardinality of a union is at most the sum of the two cardinalities, so
$$
|(\mathcal H_1\cup\mathcal H_2)(x_{1:n})|
\leq s(\mathcal H_1,n)+s(\mathcal H_2,n).
$$
Taking the supremum over samples proves the <shattering coefficient of a union> bound
$$
\boxed{s(\mathcal H_1\cup\mathcal H_2,n)
\leq s(\mathcal H_1,n)+s(\mathcal H_2,n).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 31J / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 31J / b / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 31J / b / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)