The phrase "set task" refers to the act of assigning a specific job or duty to someone. There are numerous synonyms that can be used to express this concept, depending on the context or tone of the situation. Some alternatives include "assign a task," "delegate a responsibility," "allocate a duty," "designate a mission," "entrust a chore," "impose an obligation," "appoint a charge," "set a challenge," "prescribe a project," "specify an assignment," or "enjoin a duty." Choosing the most appropriate synonym will depend on the desired level of formality, the nature of the work, and the relationship between the assigner and the assignee.