CS 300500N Introduction to Discrete Structures

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CS 300/500NIntroduction to Discrete Structures
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Discrete mathematics deals with

• Separated or discrete sets of objects
• (rather than continuous sets)
• Processes with a sequence of
• individual steps
• (rather than continuously changing
  processes)

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Goals of this class

• Study of standard facts of discrete mathematics
• Development of mathematical reasoning skills
• Discussion of applications

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Outline of Topics

• Mathematical Logic (Ch. 1,2)
• Proofs and Induction (Ch. 3,4)
• Graph Theory (Ch. 11)
• Set Theory (Ch. 5)
• Relations (Ch. 10)
• Counting Techniques (Ch. 6)
• Algorithms and Their Analysis (Ch. 9)

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Logic

• Logic is study of abstract reasoning,
• specifically,
• concerned with whether reasoning is
  correct.
• Logic focuses on
• relationship among statements
• as opposed to
• the content of any particular statement.

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Example

• Sequence of statements
• All students take CS300.
• Anyone who takes CS300 is CS major.
• Therefore, all students are CS majors.
• If (1) and (2) were true,
• then logic would assure that (3) is true.

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Outline of logic topics

• Simple Statements
• Compound Statements
• Conditional Statements
• Quantified Statements
• Valid and Invalid Arguments for all
• kind of statements

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Logical Statements

• Definition A statement is a sentence that
  is true or false but not both.
• Examples 358 (true statement)
• Today is Friday (false statement)
• Note xgty is not a statement

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Logical Connectives

• For given statements p and q
• Negation of p p (not p)
• Conjunction of p and q ( p
  and q)
• Disjunction of p and q (p
  or q)

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Truth table for negation
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Truth table for conjunction
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Truth table for disjunction
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Statement form

• Expression made up of
• statement variables (such as p,q)
• and logical connectives
• becomes a statement when
• actual statements are substituted
• for the variables.
• Example
• (Exclusive Or)

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Truth Table for a Statement Form

• Ex Truth table for

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Logical equivalence

• Statements P and Q are
• logically equivalent
• if and only if
• they have identical truth values
• for each substitution of
• their component statement variables.
• Ex

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Verifying logical equivalence

• Ex

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Important Logical Equivalences

• Double negation
• De Morgans laws
• Ex negation of -5 lt x lt 7 is

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Tautologies and Contradictions

• Tautology is a statement form
• which is true
• for all values of statement variables.
• E.g., is a tautology
• Contradiction is a statement form
• which is false
• for all values of statement variables.
• E.g., is a contradiction

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More Logical Equivalences

• Commutative laws
• Associative laws
• Distributive laws
• Absorption laws

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Simplifying Statement Forms