For set problems: (1) Draw a Venn diagram when possible. (2) Use inclusion-exclusion for counting union elements. (3) Apply De Morgan's laws to simplify complement expressions. For relation problems: (1) Check each property (reflexive, symmetric, transitive) systematically for all pairs, not just examples. (2) For equivalence relations, identify the partition and count equivalence classes. (3) Remember that a relation can be both symmetric and antisymmetric (e.g., the identity relation). For function problems: (1) Verify the definition — every element of domain must map to exactly one element of codomain. (2) For injectivity, check if the function is monotonic. (3) For surjectivity, check if every codomain element has a preimage. (4) For counting, choose the appropriate formula ($n^m$, P(n,m), inclusion-exclusion, or n!).