MMW NOTES

Proposition

A declarative sentence that can be objectively identified as either true or false.

Truth Table

A diagram in table form that shows all the possible truth values of a proposition.

Universal quantifiers

Terms such as "all, each, every, no, none" that indicate a quantified statement.

Existential quantifiers

Terms such as "some, there exists, at least one" that indicate a quantified statement.

Negation

The proposition that is false when the original proposition is true, and true when the original proposition is false.

Conjunction (^)

The compound proposition that is true only when both p and q are true

Disjunction (∨)

compound proposition that is false only when both p and q are false and true otherwise

Exclusive or

The compound proposition that is true when exactly one of the two propositions is true, and false otherwise.

Conditional statement

false when p is true and q is false (T → F) and true otherwise

Biconditional statement

true only when p and q have the same truth value

Tautology

A compound proposition that is always true, regardless of the truth values of its variables.

Contradiction

A compound proposition that is always false, regardless of the truth values of its variables.

Logically implies

• The relationship between two propositions where the conditional statement p→q is a tautology.

• denoted by p=>q

Logically equivalent

• if p=>q and q=>p, then they are logically equivalent

• denoted as p<=>q

Contingency

A compound proposition that is neither a tautology nor a contradiction.

De Morgan's Laws

The laws that describe the negation of conjunction and disjunction.

Valid argument

An argument where if all the premises are true, then the conclusion must be true.

Invalid argument

An argument where the truth of the premises does not guarantee the truth of the conclusion.

Fallacy

An error in reasoning that leads to an invalid argument.

Euler Diagram

A visual representation used to analyze arguments based on the relationships between sets.

Inductive Reasoning

The process of reasoning that arrives at a general conclusion based on the observation of specific examples.

Specimen

An object defined by a premise.

Conjecture

The generalization made in an inductive reasoning process.

Counterexample

A specimen that negates the conjecture made in an inductive reasoning process.

Strong Inductive Argument

An inductive argument that makes a compelling case for its conclusion.

Weak Inductive Argument

An inductive argument whose conclusion is not well supported by the premises.

Deductive Reasoning

The process of reasoning that arrives at a conclusion based on previously accepted general statements.

Axioms

Basic true statements used in deductive reasoning.

Theorems

Derived true statements from axioms.

Valid Deductive Statement

A deductive statement where the conclusion follows from its premises, regardless of the truth of the premises or conclusion.

Sound Deductive Statements

Deductive statements that are valid and have all true premises.

George Polya

Mathematician who devised a model for problem solving in his book "How to Solve It".

Heuristic

Polya's problem-solving model, serving to discover solutions.

Understand the problem

The first step in Polya's problem-solving model, involving asking questions, experimenting, or replacing the question with your own words.

Determine what is asked

identifying the desired outcome.

Determine the given facts

, identifying the provided information.

Have initial illustrations to visualize the setting

Creating visual representations to better understand the problem.

Devise a plan

The second step in Polya's problem-solving model, finding the connection between the data and the unknown.

Carry out the plan

third step; Implementing the chosen operation, procedure, or formula.

Look back

The final step in Polya's problem-solving model, examining the obtained solution and checking its reasonableness.