EECS 203
Chapter 1 - Foundations: Logic and Proofs
1.1 Propositional Logic
Proposition
Truth Table(2^n)
Connectives
Negation
Conjunction
Disjunction
Implication
Converse
Inverse
Contrapositive(equivalent)
Operator Precedence
1.2 Applications of Propositional Logic
System Specifications
Consistent
Contradiction
Solving Logic Puzzles
1.3
Logical Equivalence
Tautology
Contradiction
Contingency
De Morgan's Law
Disjunctive Normal Form
Boolean Logic Laws
Identity
Domination
Idempotent
Double Negation
Commutative
Associative
Distributive
De Morgan's
Absorption
Negation
Propositional Satisfiability
Satisfiable
Unsatisfiable
Solution
Solving Satisfiability Problems(w/o truth table)
1.4 Predicates and Quantifiers
Predicate Logic
Predicate
Propositional Function P(x)
Preconditions - Valid Input
Postconditions - Valid Outputs
Quantifiers
Quantification
All, some, many, none, few
Domain of Discourse
Universal Quantifier(For All)
Counterexample
Existentital Quantifier(There exists)
Uniqueness Quantifier
Quantifier Precedence
Binding Variables
Logical Equivalence Involving Quantifiers
Negating Quantified Expressions(De Morgan's)
Translating from English
Quantifiers in System Specifications
1.5 Nested Quantifiers
Quantification as Loops
Order of Quantifiers
Scope
Multiplicative Inverse
Translating to/from English
Negating Nested Quantifiers
1.6 Rules of Inference
Argument
Premise
Valid
Fallacies
Conclusion
Argument Forms
Modus ponens(law of detachment)
Modus tollens
Hypothetical syllogism
Disjunctive syllogism
Addition
Simplification
Conjunction
Resolution
Building Arguments
Common Fallacies
Fallacy of affirming the conclusion
Fallacy of denying the hypothesis
Universal Instantiation
Universal Generalization
Existential Instantiation
Existential Generalization
Combining rules of inference for propositions and quantified statements
1.7 Introduction to Proofs
Informal Proofs
Proof
Theorem
Axioms(Postulates)
Lemma
Corollary
Conjecture
Omitting Quantifiers
Methods of Proving Theorems
Direct Proof
Parity
Proof by Contraposition
Indirect Proof
Proof by Contradiction
Proof of Equivalence
Counterexamples
Common Mistakes in Proofs
Vacuous and Trivial Proofs
"begging the question"
"circular reasoning"
Definitions:
Integer
Even
Odd
Perfect square
Sum of all integers less than n
Rational number
General Proof Strategy
Readability(compassion)
1.8 Proof Methods and Strategy
Exhaustive Proof
Proof by Cases
Leveraging Proof by Cases
Without Loss of Generality
Common Errors with Exhaustive Proofs and Proof by Cases
Existence Proofs
Witness
Constructive
Non-constructive
Uniqueness Proofs
Proof Strategies
Forward and backward reasoning
Adapting Existing Proofs
Looking for Counterexamples
Tiling Problems
Definitions:
Perfect Power
Arithmetic Mean
Metric Mean
Fermat's Theorem
Chapter 2 Basic Structures: Sets, Functions, Sequences, Sums, and Matrices
2.1
2.2
2.3
2.4
2.5
2.6
Chapter 5
5.1
5.2
1.1 Propositional Logic
Proposition
Truth Table(2^n)
Connectives
Negation
Conjunction
Disjunction
Implication
Converse
Inverse
Contrapositive(equivalent)
Operator Precedence
1.2 Applications of Propositional Logic
System Specifications
Consistent
Contradiction
Solving Logic Puzzles
1.3
Logical Equivalence
Tautology
Contradiction
Contingency
De Morgan's Law
Disjunctive Normal Form
Boolean Logic Laws
Identity
Domination
Idempotent
Double Negation
Commutative
Associative
Distributive
De Morgan's
Absorption
Negation
Propositional Satisfiability
Satisfiable
Unsatisfiable
Solution
Solving Satisfiability Problems(w/o truth table)
1.4 Predicates and Quantifiers
Predicate Logic
Predicate
Propositional Function P(x)
Preconditions - Valid Input
Postconditions - Valid Outputs
Quantifiers
Quantification
All, some, many, none, few
Domain of Discourse
Universal Quantifier(For All)
Counterexample
Existentital Quantifier(There exists)
Uniqueness Quantifier
Quantifier Precedence
Binding Variables
Logical Equivalence Involving Quantifiers
Negating Quantified Expressions(De Morgan's)
Translating from English
Quantifiers in System Specifications
1.5 Nested Quantifiers
Quantification as Loops
Order of Quantifiers
Scope
Multiplicative Inverse
Translating to/from English
Negating Nested Quantifiers
1.6 Rules of Inference
Argument
Premise
Valid
Fallacies
Conclusion
Argument Forms
Modus ponens(law of detachment)
Modus tollens
Hypothetical syllogism
Disjunctive syllogism
Addition
Simplification
Conjunction
Resolution
Building Arguments
Common Fallacies
Fallacy of affirming the conclusion
Fallacy of denying the hypothesis
Universal Instantiation
Universal Generalization
Existential Instantiation
Existential Generalization
Combining rules of inference for propositions and quantified statements
1.7 Introduction to Proofs
Informal Proofs
Proof
Theorem
Axioms(Postulates)
Lemma
Corollary
Conjecture
Omitting Quantifiers
Methods of Proving Theorems
Direct Proof
Parity
Proof by Contraposition
Indirect Proof
Proof by Contradiction
Proof of Equivalence
Counterexamples
Common Mistakes in Proofs
Vacuous and Trivial Proofs
"begging the question"
"circular reasoning"
Definitions:
Integer
Even
Odd
Perfect square
Sum of all integers less than n
Rational number
General Proof Strategy
Readability(compassion)
1.8 Proof Methods and Strategy
Exhaustive Proof
Proof by Cases
Leveraging Proof by Cases
Without Loss of Generality
Common Errors with Exhaustive Proofs and Proof by Cases
Existence Proofs
Witness
Constructive
Non-constructive
Uniqueness Proofs
Proof Strategies
Forward and backward reasoning
Adapting Existing Proofs
Looking for Counterexamples
Tiling Problems
Definitions:
Perfect Power
Arithmetic Mean
Metric Mean
Fermat's Theorem
Chapter 2 Basic Structures: Sets, Functions, Sequences, Sums, and Matrices
2.1
2.2
2.3
2.4
2.5
2.6
Chapter 5
5.1
5.2