HSSLIVE Plus One Maths Chapter 14: Mathematical Reasoning Notes

This chapter develops students’ abilities to construct valid arguments and evaluate logical claims. Students will study propositions, connectives, quantifiers, and methods of proof including direct proof, proof by contradiction, and proof by cases. By strengthening analytical thinking and precision in communication, mathematical reasoning equips students with skills that extend beyond mathematics into areas requiring critical thinking, computer programming, legal analysis, and scientific research. This foundation in logic helps students approach complex problems systematically.

Chapter 14: Mathematical Reasoning

Statements:

A statement is a declarative sentence that is either true or false, but not both.

Types of Statements:

1. Simple Statement: Contains a single assertion
2. Compound Statement: Formed by connecting simple statements using logical connectives

Logical Connectives:

1. Conjunction (∧): “and” operation
   • p ∧ q is true only when both p and q are true
2. Disjunction (∨): “or” operation
   • p ∨ q is true when either p or q (or both) is true
3. Negation (¬): “not” operation
   • ¬p has the opposite truth value of p
4. Implication (→): “if…then” operation
   • p → q is false only when p is true and q is false
5. Biconditional (↔): “if and only if” operation
   • p ↔ q is true when p and q have the same truth value

Tautology and Contradiction:

• Tautology: A compound statement that is always true
• Contradiction: A compound statement that is always false

Quantifiers:

1. Universal Quantifier (∀): “for all”
2. Existential Quantifier (∃): “there exists”

Methods of Proof:

1. Direct Proof: Start with known facts and directly deduce the conclusion
2. Proof by Contradiction: Assume the negation of the statement and derive a contradiction
3. Proof by Contrapositive: Prove the contrapositive statement (¬q → ¬p) instead of (p → q)
4. Mathematical Induction: Involves a base case and an inductive step