Mathematical reasoning & logic

MATHEMATICAL REASONING (CHAPTER 4 FORM 4)	LOGIC ( CHAPTER 1 TMC 2013 )
Statement A sentences that is either true or false ØDetermine whether the given sentence is a statement. ØTo determine whether the statements are true or false. ØConstruct true or false statements using given numbers and mathematics symbols.	Proposition A declarative statements that either true or false ØTo determine whether the propositions are true or false.

Quantifiers ·All ·Some ØConstruct statements using quantifiers. ØDetermine whether the statements contain quantifiers are true or false. ØDetermine quantifiers can be generalized to cover all cases by using quantifier ‘all’. ØConstruct a true statement using quantifiers.

Predicates and Quantifiers ·universal quantifiers ["x P(x)] ·existential quantifiers [$x P(x)] ØIdentifying predicates. ØDetermine true or false predicates. ØQuantifier negation ØMultiple quantifiers.

Operations on statement Negation of a statement ØChange the truth value of given statement by placing the word not or no Using symbols : ~ Compound statement : AND and OR ØIdentify two statements from compound statements. ØForm a compound statement by combining two given statements using word ‘and’ and ‘or’ ØDetermine the truth value of a compound statement which is the combination of two statements.

Logical operators Negation ØSymbol of negation : ¬ Logical operators. Conjunction [AND (Ù)], disjunction [OR (Ú)], exclusive OR [ XOR (⊕)] ØTo construct the truth tables based on logical operators. Precedence of logical operators Negation (¬), conjunction (Ù), disjunction (Ú), implication (→), biconditional (↔). Propositional equivalence ØLogical equivalences if pΞq is tautology ØProving using logic laws

Implication Antecedent and consequent of an implication. ØIdentify the antecedent and consequent of an implication ‘if p, then q’. ØConstruct the mathematical statements in the form of if p then q. Combining two implications using ‘if and only if’ ØWrite two implications from a compound statement ØConstruct a mathematical statements Converse of an implication ØDetermine the converse of a given implication ØDetermine whether the implication is true or false

Logical connective : implication Biconditional statement (↔) ØConstruct the truth table. ØTo prove about tautology Converse, inverse, contrapositive ØDetermine the converse, inverse, contrapositive of a given proposition.

Argument Premise and conclusion of an argument. ØIdentify the premises and conclusion of a given simple argument ØDraw a conclusion based on two given premise and vise versa. Argument form ØMake a conclusion based on ·argument I ·argument II ·argument III ØComplete an argument, given a premise and the conclusion.

Rules of inference ØPattern of logically valid deductions from hypotheses to conclusion ØProving using all rules of inferences. ØTo produce a valid argument. There are: ·Hypothetical syllogism ·Modus ponens ·Modus tollen ·Addition ·Simplification ·Conjunction ·Disjunctive syllogism ·Resolution vThere are various forms of incorrect reasoning called fallacies that lead to invalid argument. (Affirming the conclusion and denying the hypotheses.

DEDUCTION AND INDUCTION •Deductive and inductive reasoning. •Conclusion by deductive reasoning. •Conclusion by inductive reasoning.

Proof techniques Lemma, corollary, conjecture ØDirect proof ØIndirect proof ØTrivial proof ØVacuous proof ØProof by contradiction.