Sets
Sets tests representation, subsets, power sets, set operations, Venn diagrams, and cardinality. The exam often uses a word problem with two or three overlapping categories.
Core rules
A set is a well-defined collection; roster form lists elements, set-builder form gives a rule. A⊆B means every element of A is in B. The power set P(A) has 2ⁿ elements if A has n elements. Operations: A∪B, A∩B, A−B, and complement A′ relative to the universal set. De Morgan’s laws: (A∪B)′=A′∩B′ and (A∩B)′=A′∪B′. Cardinality: n(A∪B)=n(A)+n(B)−n(A∩B). For three sets, add singles, subtract pairwise intersections, add the triple intersection. Disjoint sets have empty intersection.
Worked example
In a class of 40, 25 like mathematics, 20 like science, and 10 like both. Then n(M∪S)=25+20−10=35. Students liking neither=40−35=5.
Traps
The empty set is a subset of every set, but it is not an element of every set. Distinguish ∈ from ⊆. n(A−B)=n(A)−n(A∩B), not n(A)−n(B). In three-set Venn problems, fill the triple intersection first. The complement depends on the stated universal set; do not assume it is the set of real numbers.