# How many possible hypotheses

The training set contains $$p$$ papers. Each paper is annotated as research or non-research. To develop the research paper filter, we consider the $$W$$ most frequent phrases in a paper. The research paper filter will use the presence/absence of these $$W$$ phrases to decide if the paper is indeed a research paper or not.

1. How many possible hypothesis are there?

I have $$2^W$$ possible hypothesis, but if we are including conjunctive hypothesis symbols, then I would say $$4^W$$. Is my logic correct?

2. How many of them are consistent?

For this question, I have no idea where to begin. Any insight?

I would first create a definition of possible hypotheses (i.e., $H_{0}$ and $H_{1}$ are my two possible hypotheses, and every hypothesis will have both a null and an alternative), and then associate a $H_{0}$ with a $H_{1}$ such that a combination of both null and alternative hypotheses equate to one hypothesis.
• With $p \in P$, where $p$ is a paper in the set of papers $P$,
• With $w \in W$, where $w$ is a word in a set of words $W$ that indicate whether $p$ is a research paper or not,
• With $i$ as the number of times a $w$ occurred in a $p$,
• count the number of times a $w$ occurs in each $p$ and iterate $i$ accordingly.