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1 CS-227, Discrete Structures I Spring 2006 Semester Summary of Course Coverage 1) Propositional Calculus a) Negation (logical NOT) b) Conjunction (logical AND) c) Disjunction (logical inclusive-or) d) Inequalities e) Propositional Form f) Truth Table g) Logical Equivalence h) Double Negative i) Negation of AND and of OR: DeMorgan s Laws and their application j) Inequalities and DeMorgan s Laws k) Tautology l) Contradiction m) Logical Equivalences (NOTE: This is summarized in Theorem on p. 14.) n) Simplification of Logical Statements o) Conditional Statement: Hypothesis and Conclusion p) Vacuous Truth (Truth by Default) q) Order of Precedence of Logical Operators r) Negation of a Conditional Statement s) Forms Related to a Conditional Statement i) Converse ii) Inverse iii) Contrapositive t) Biconditional (Only-If) u) Necessary Conditions v) Sufficient Conditions (continued) Page 1 of 6

2 2) Forms of Arguments: either valid or invalid a) Premises/Hypotheses/Assumptions, and Conclusions b) Valid Forms of Argument (NOTE: These are summarized in Table on p. 40.) i) Modus Ponens (Method of Affirming) ii) Modus Tollens (Method of Denying) iii) Generalization iv) Specialization v) Transitivity vi) Division into Cases c) Fallacies or Invalid Forms of Argument i) Using Ambiguous Premises ii) Begging the Question iii) Jumping to a Conclusion iv) Converse Error v) Inverse Error 3) Digital Logic Circuits a) Buffer Gate: Logical Concept, Standard for Drawing, Standard for Labeling b) NOT Gate: Logical Concept, Standard for Drawing, Standard for Labeling c) AND Gate: Logical Concept, Standard for Drawing, Standard for Labeling d) [Inclusive-] OR GATE: Logical Concept, Standard for Drawing, Standard for Labeling e) Relationship between Truth Table, Boolean Expression, and Digital Logic Circuit i) [NOT-] AND-OR ( Sum of Products ) Circuit ii) [NOT-] OR-AND ( Product of Sums ) Circuit f) NAND Gate and its logical equivalent, the Invert-OR Gate: Logical Concept, Standard for Drawing, Standard for Labeling g) NOR Gate and its logical equivalent, the Invert-AND Gate: Logical Concept, Standard for Drawing, Standard for Labeling h) Straightforward Simplification of Boolean Logic Circuits i) NAND-NAND equivalent to each and every Sum of Products Circuit ii) NOR-NOR equivalent to each and every Product of Sums Ciircuit 4) Representation of Numbers a) Positional Representational Schemes for Number Representation i) Radix R ii) Numerals: 0 (R 1) iii) Radix Point to separate integer part of the number from fractional part b) Decimal Number Representation (generalized Radix Point becomes the Decimal Point) c) Binary Number Representation (generalized Radix Point becomes the Binary Point): Non-Negative Numbers and Numbers of Mixed Sign Page 2 of 6

3 i) Unsigned (i.e., Non-Explicitly-Signed) Numbers ii) Signed-Magnitude iii) Ones -Complement iv) Two s-complement d) Inter-Conversion of Binary, Octal, and Hexadecimal Number Representations (NOTE that generalized Radix Point becomes Octal Point or Hexadecimal Point for those representations) e) Use of Hexadecimal Notation to Describe the Contents of a Register or of a Memory Location f) Binary Addition i) The Half-Adder ii) The Full-Adder g) The Four Processor-Status Bits, their Meaning and Significance: i) C bit ii) Z bit iii) N bit iv) V bit h) Specification of the Content of Each Status Bit for each of the Schemes of Binary Number Representation i) C bit: What, if anything, does it signify? ii) Z bit: How many Zeroes? iii) N bit iv) V bit: Recognizing the occurrence of overflow. 5) Predicate Calculus a) Definition of Predicate b) Domain of a Predicate Variable i) The Set of [Mathematically] Real Numbers: R ii) The Set of Integers: Z, Z +, Z, Z nonneg iii) The Set of Rational Numbers (i.e., numbers expressible as Quotients of two integers): Q c) The Truth Set of a Predicate d) The Universal Qualifier: e) The Existential Qualifier: f) Equivalence of Truth Sets: and g) Negation of a Universal Statement h) Negation of an Existential Statement i) Relationships among:,,, and j) Negation of a Universal Conditional Statement k) Vacuous Truth l) A Predicate and its Inverse, Converse, and Contrapositive m) Necessary Conditions, Sufficient Conditions, and Only-If in Quantified Statements Page 3 of 6

4 6) Elementary Number Theory a) Even Integers: Formal Definition b) Odd Integers: Formal Definition c) Prime Numbers: Formal Definition d) Composite Numbers: Formal Definition e) Constructive Proof of Existence: Find an Example f) Disproof of a Universal Statement by means of a Counterexample g) Rules for Writing Formal Proofs: described in detail on page 134 h) Common Mistakes made in the course of attempting a proof: i) Generalization from an example ii) Absent-Minded Use of a Single Variable in Two or More Definitions iii) Jumping to a Premature Conclusion iv) Begging the Question i) Rationality of the Sum of Two Rational Numbers j) Rationality of the Difference between Two Rational Numbers k) Rationality of the Product of Two Rational Numbers l) Retention of Integer Property under Addition, Subtraction, and Multiplication (but not under division) m) Divisibility n) Fundamental Theorem of Arithmetic (a.k.a. the Unique Factorization Theorem) Mid-Term coverage ended here. o) Quotient-Remainder Theorem p) Types of Proofs I: Direct Proof q) Types of Proofs II: Disproof via Counterexample r) Types of Proofs III: Division Into Cases s) Types of Proofs IV: Contradiction t) Types of Proofs V: Contraposition u) Types of Proofs VI: Simple Mathematical Induction v) Types of Proofs VII: Strong Mathematical Induction w) Binary Representation of an Integer x) Parity y) Opposite Parity of Consecutive Integers z) Floor Function aa) Ceiling Function bb) Non-Existence of a Greatest Integer cc) Mutual Exclusivity of the Properties Even and Odd dd) Mixed Addition of Rational and Irrational Numbers Page 4 of 6

5 ee) Irrationality of Square Root of 2 ff) Infinitude of the Set of Prime Numbers 7) Algorithms a) Finding the Greatest Common Divisor: the Euclidean Algorithm b) Conversion from Decimal Representation to Another Radix I: Integer Part c) Conversion from Decimal Representation to Another Radix II: Fractional Part d) Pre-Conditions for Loops e) Post-Conditions for Loops f) The Loop Invariant g) Inevitability of Exit from the Loop h) Correctness of a Loop to Compute a Product i) Correctness of the Division Algorithm j) Correctness of the Euclidian Algorithm 8) Sequences a) Primitive Specification of terms: Enumeration b) Symbolic Specification of Terms: Explicit Formula c) Summation Notation vs. Expanded Notation d) Index of Summation, Lower Limit, Upper Limit e) Computing a Summation f) Interchange between Expanded Form and Summation Notation g) Separating Off of a Final Term h) Adding On of a Final Term i) Product Notation j) Computing a Product k) Properties of Summations and of Products l) Change of Variables m) Factorial Notation n) Computing Factorials o) Sum of First n Consecutive [Positive/Non-Negative] Integers p) Sum of a Geometric Sequence q) A Property of Divisibility Proved by Mathematical Induction r) An Inequality Proved by Mathematical Induction s) The Well-Ordering Principle of Integers t) Divisibility of Every Integer by a Prime Number u) Quotient-Remainder Theorem 9) Set Theory a) Definition I: Set Page 5 of 6

6 b) Definition II: Element c) Irrelevance of the Order in which the elements are listed d) Irrelevance of Multiple Listings of a single element e) Notation: {} f) Specification of Set Contents via Enumeration g) Specification of Set Contents via Property Definition h) Subset: Definition i) Proper Subset: Definition j) Set Equality k) Venn Diagrams l) Set Operations I: Union m) Set Operations II: Intersection n) Set Operations III: Difference o) Set Operations IV: Complementation c p) The Empty Set (a.k.a. Null Set): {} or q) Disjoint Sets: Mutually Disjoint, Pairwise Disjoint r) Partition of a Set s) Power Set t) The Ordered Pair u) Cartesian Product v) Set Identities: Analogous to Logical Identities Page 6 of 6

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