Mathematical Logic and Set Theory
This domain carries short, high-accuracy marks: truth values of compound statements, truth tables, logical equivalence and duality, plus counting problems on sets and Venn diagrams. Questions are usually small, so a single sign or symbol error costs the whole question.
Statements and connectives
- A statement is an assertive sentence with exactly one truth value, T (1) or F (0). Questions, orders, exclamations and open sentences like x + 3 = 7 are not statements.
- Negation ~p reverses the value. Conjunction p∧q is true only when both are true; disjunction p∨q is false only when both are false; implication p→q is false only when p is true and q is false; biconditional p↔q is true when the values agree.
- p→q ≡ ~p∨q ≡ ~q→~p. The converse is q→p and the inverse is ~p→~q; only the contrapositive is equivalent to the original.
- Duality: interchange ∧ with ∨ and T with F, leaving negations unchanged.
- De Morgan: ~(p∧q) ≡ ~p∨~q and ~(p∨q) ≡ ~p∧~q.
Sets and counting
n(A∪B) = n(A) + n(B) − n(A∩B). For three sets, n(A∪B∪C) = sum of singles − sum of pairs + n(A∩B∩C). "At least one" means union; "exactly one" means union minus intersection.
Worked example
In a class of 60, 35 like Mathematics, 30 like Statistics and 15 like both. n(A∪B) = 35 + 30 − 15 = 50 like at least one. Neither = 60 − 50 = 10. Exactly one = 50 − 15 = 35, which also equals (35 − 15) + (30 − 15).
Traps
- Offering x is a prime number as a statement - it has no fixed truth value.
- Confusing the converse with the contrapositive.
- The negation of all students passed is at least one student failed.
- Forgetting to subtract the both group once in counting problems.