For any two documents x and z, define k (x, z) to equal the number of unique words that occur in both x and z (i. e., the size of the intersection of the sets of words in the two documents). Is this function a kernel?

Yes, the function k (x, z) is a kernel

Explanation:

To show that k (x, z) is a kernel.

First, we'll need to explicitly construct two features of k (x, z).

The features are:

vector φ (x) and vector φ (z) in such a way that K (x, z) = φ (x) ·φ (z).

Then, we proceed to the next step ...

For any given documents, a vocabulary V can be constructed.

Vocabulary V is finite size for the words in the document set.

Given V, a feature mapping φ (x) can then be constructed for x by the following:

For the kth word wk in V, if wk appears in document x, assign φ (x) k (the kth element of φ (x)) to be 1; else assign 0 to it.

Then the number of unique words common in x and z is φ (x) ·φ (z).

This gives us the kernel.