Sets and Logic
This domain tests set notation, Venn-diagram counting, and the truth conditions of compound statements, including the conditional and its relatives.
Set vocabulary and operations
- Roster form {2, 4, 6}; set-builder {x | x is even and 0 < x < 8}.
- Element versus subset: 2 ∈ {2, 4}, but {2} ⊆ {2, 4}. The empty set ∅ is a subset of every set.
- Cardinality |A| is the number of elements; a set with n elements has 2ⁿ subsets.
- Union A ∪ B: everything in A or B or both. Intersection A ∩ B: only elements in both. Complement A′: elements of the universal set U that are not in A.
- Disjoint sets have an empty intersection.
For two sets, |A ∪ B| = |A| + |B| − |A ∩ B|, and the number in neither set is |U| − |A ∪ B|. The subtraction corrects for elements counted twice.
Logic
- Negation ~p reverses truth value.
- Conjunction p ∧ q is true only when both parts are true.
- Disjunction p ∨ q is true when at least one part is true (inclusive or).
- Conditional p → q is false in exactly one case: p true and q false.
- Related statements: converse q → p; inverse ~p → ~q; contrapositive ~q → ~p. Only the contrapositive is logically equivalent to the original.
- De Morgan's laws: ~(p ∧ q) ≡ ~p ∨ ~q and ~(p ∨ q) ≡ ~p ∧ ~q.
Worked example
In a survey of 40 students, 25 like coffee, 18 like tea, and 10 like both.
- Coffee or tea: 25 + 18 − 10 = 33.
- Neither: 40 − 33 = 7.
- Coffee only: 25 − 10 = 15. Tea only: 18 − 10 = 8.
- Check: 15 + 8 + 10 + 7 = 40, so the regions are consistent.
Traps that cost marks
- Forgetting to subtract the overlap, which double-counts the "both" group.
- Treating the converse as equivalent to the original; "if it rains, the ground is wet" does not license "the ground is wet, so it rained."
- Reading "or" as exclusive when the statement does not say so.
- Mishandling the negation of a conditional: ~(p → q) is p ∧ ~q, not ~p → ~q.
- Computing a complement relative to the wrong universal set.