In this paper, we investigated semantic communication for multi-task processing using an information-theoretic approach. We introduced the concept of a "semantic source", allowing multiple semantic interpretations from a single observation. We formulated an end-to-end optimization problem taking into account the communication channel, maximizing mutual information (infomax) to design the semantic encoding and decoding process exploiting the statistical relations between semantic variables. To solve the problem we perform data-driven deep learning employing variational approximation techniques. Our semantic encoder is divided into a common unit and multiple specific units to facilitate cooperative multi-task processing. Simulation results demonstrate the effectiveness of our proposed semantic source and system design when statistical relationships exist, comparing cooperative task processing with independent task processing. However, our findings highlight that cooperative multi-tasking is not always beneficial, emphasizing the importance of statistical relationships between tasks and indicating the need for further investigation into the semantically processing of multiple tasks.