Sets
Sets introduces the language of collections and membership. The exam tests notation, subsets, operations, Venn diagrams and the cardinality formula for two or three sets.
Core rules
- A set is a well-defined collection; order and repetition do not matter.
- A⊆B means every element of A is in B; the empty set ∅ is a subset of every set; a set with n elements has 2ⁿ subsets.
- Operations: A∪B (union), A∩B (intersection), A−B (difference), A′ (complement in the universal set).
- De Morgan: (A∪B)′=A′∩B′; (A∩B)′=A′∪B′.
- For two finite sets: n(A∪B)=n(A)+n(B)−n(A∩B).
Worked example
In a class of 40, 25 play football, 18 play cricket and 10 play both. Then n(F∪C)=25+18−10=33, so 40−33=7 play neither.
Traps
Do not count the intersection twice; subtract it once. Distinguish ∈ (element) from ⊆ (subset). The empty set is not the same as {0}. In a Venn diagram, place shared elements in the overlap before filling the only regions. For three sets, subtract pairwise intersections and add back the triple intersection.