Logic and Sets
The Logic and Sets domain combines set notation and Venn-diagram counting with truth-functional logic. You need to compute the sizes of unions, intersections, and complements, and to judge whether a compound statement or an argument form is valid. Many questions are solved fastest by sketching a two-circle Venn diagram and filling regions from the inside out.
Set operations
- Union A∪B contains everything in either set; intersection A∩B contains only shared elements; complement A′ contains elements of the universal set U outside A; difference A − B contains elements in A but not B.
- Inclusion–exclusion: |A∪B| = |A| + |B| − |A∩B|.
- Three sets: |A∪B∪C| = Σ|A| − Σ|A∩B| + |A∩B∩C|.
- "Neither" = |U| − |A∪B|; "only A" = |A| − |A∩B|.
Truth and conditionals
- A conditional p→q is false only when p is true and q is false.
- The converse q→p and the inverse ~p→~q are not equivalent to the original statement; the contrapositive ~q→~p is equivalent.
- De Morgan: ~(p∧q) ≡ ~p∨~q and ~(p∨q) ≡ ~p∧~q.
- A biconditional p↔q is true exactly when p and q have the same truth value.
Valid argument forms
- Modus ponens: p→q and p, therefore q.
- Modus tollens: p→q and ~q, therefore ~p.
- Affirming the consequent (p→q and q, therefore p) is invalid.
Worked example
Of 40 students, 25 take math, 18 take physics, and 10 take both. How many take neither? |M∪P| = 25 + 18 − 10 = 33. Neither = 40 − 33 = 7. Students taking math only = 25 − 10 = 15, and physics only = 18 − 10 = 8. Check: 15 + 8 + 10 + 7 = 40.
Traps
- Adding |A| and |B| without subtracting the overlap, which double-counts the shared members.
- Treating the converse of a statement as if it were equivalent to the original.
- Reading everyday "or" as exclusive when logic uses inclusive or, so p∨q is true when both hold.
- Taking a complement without a defined universal set, which leaves the answer undefined.