This study focuses on the relationships established between the elements that compose the community of inquiry (CoI): cognitive, social, and teaching presence. Using three questionnaires, we analyse the students' perception of synchronous and asynchronous virtual in text-based communication (chats, forums and emails). Starting from the high correlations found between the three elements, we perform a multiple linear regression analysis. The findings show that relationships can be established in the model in which the cognitive elements are strongly predicted, to a greater extent by social presence than by teaching presence. In the forums the cognitive presence is better explained by other presences than in chats and emails. The results reveal the need to analyse the three kinds of presence jointly, assessing the impact of each on student learning. We also determine that the instructor benefits from knowing which tool is more valid for the learning objectives.