Sets
What this domain tests
Sets tests notation, operations and counting. Expect roster-to-set-builder conversions, two-set Venn diagrams, and direct use of the cardinality formula n(A∪B) = n(A) + n(B) − n(A∩B).
Core rules
- Roster form lists elements; set-builder form states a property. The empty set ∅ is a subset of every set.
- A set with n elements has 2^n subsets, including ∅ and the set itself; it has 2^n − 1 proper subsets.
- A∪B contains elements in A or B; A∩B contains elements in both; A−B contains elements in A but not B.
- Complement A′ means elements of the universal set ξ that are not in A.
- De Morgan's laws: (A∪B)′ = A′∩B′ and (A∩B)′ = A′∪B′.
- Useful counts: n(A−B) = n(A) − n(A∩B), and n(A′) = n(ξ) − n(A).
Worked example
In a class of 40, 25 like Mathematics, 20 like Science and 10 like both. Find the number who like neither. By the union formula, n(M∪S) = 25 + 20 − 10 = 35. Neither = 40 − 35 = 5. Also, only Mathematics = 25 − 10 = 15; only Science = 20 − 10 = 10. Checking: 15 + 10 + 10 + 5 = 40.
Traps
- Counting the intersection twice when adding the two sets.
- Mixing ∈ (element of) with ⊂ (subset of).
- Forgetting that ∅ and the set itself are subsets when counting a power set.
- Taking a complement without a stated universal set.
- Reading “only A” as n(A) instead of n(A) − n(A∩B).