What is the relationship between AI and data science?

Though neither are well defined, as commonly used they are somewhat orthogonal concepts.

In my opinion, AI has a fairly narrow definition - it is about optimization through actions. AI is about decision making, either in deterministic or probabilistic environments. Typically, this is operationalized as action selection to maximize some reward function, or equivalently to minimize some loss function. Supervised or unsupervised learning (i.e. machine learning) about your environment can be helpful in using your experience to aid in selecting optimal actions.

As it's commonly used, data science has no rigorous definition - businesses use the term to refer to anything from creating charts Excel to deep reinforcement learning models that can win Go. From the point of view of a practitioner, these have absolutely nothing in common. From the point of view of a business owner, the common thread is extracting meaning, and therefore value, from raw data. A data scientist is a 'meaning extraction layer'. How this operation is performed, and the techniques used (again in my experience), make no difference to the employer of a data scientist. The job title may as well be 'data magician'. But the point is that data + data science = business value, whether that comes in the form of insights into customer trends, causal analysis of marketing campaigns, or an AI bot that is 'rewarded' when it recommends you a movie that you like. I suppose that means that AI is a subset of data science - but you could also say the same thing about clear communication, so it's a bit of a non-statement.