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Discrete Mathematics, Part II CSE 2353 Fall 2007. Margaret H. Dunham Department of Computer Science and Engineering Southern Methodist University Some slides provided by Dr. Eric Gossett; Bethel University; St. Paul, Minnesota
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Discrete Mathematics, Part IICSE 2353Fall 2007 • Margaret H. Dunham • Department of Computer Science and Engineering • Southern Methodist University • Some slides provided by Dr. Eric Gossett; Bethel University; St. Paul, Minnesota • Some slides are companion slides for Discrete Mathematical Structures: Theory and Applications by D.S. Malik and M.K. Sen
Outline • Introduction • Sets • Logic & Boolean Algebra • Proof Techniques • Counting Principles • Combinatorics • Relations,Functions • Graphs/Trees • Boolean Functions, Circuits
Proof Technique: Learning Objectives • Learn various proof techniques • Direct • Indirect • Contradiction • Induction • Practice writing proofs • CS: Why study proof techniques?
Proof Techniques • Theorem • Statement that can be shown to be true (under certain conditions) • Typically Stated in one of three ways • As Facts • As Implications • As Biimplications
Validity of Arguments • Proof: an argument or a proof of a theorem consists of a finite sequence of statements ending in a conclusion • Argument: a finite sequence of statements. • The final statement, , is the conclusion, and the statements are the premises of the argument. • An argument is logically valid if the statement formula is a tautology.
Proof A mathematical proof of the statement S is a sequence of logically valid statements that connect axioms, definitions, and other already validated statements into a demonstration of the correctness of S. The rules of logic and the axioms are agreed upon ahead of time. At a minimum, the axioms should be independent and consistent. The amount of detail presented should be appropriate for the intended audience.
Proof Techniques • Direct Proof or Proof by Direct Method • Proof of those theorems that can be expressed in the form ∀x (P(x) → Q(x)), D is the domain of discourse • Select a particular, but arbitrarily chosen, member a of the domain D • Show that the statement P(a) → Q(a) is true. (Assume that P(a) is true • Show that Q(a) is true • By the rule of Choose Method (Universal Generalization), ∀x (P(x) → Q(x)) is true
Proof Techniques • Indirect Proof • The implication P → Q is equivalent to the implication ( Q → P) • Therefore, in order to show that P → Q is true, one can also show that the implication ( Q → P) is true • To show that ( Q → P) is true, assume that the negation of Q is true and prove that the negation of P is true
Proof Techniques • Proof by Contradiction • Assume that the conclusion is not true and then arrive at a contradiction • Example: Prove that there are infinitely many prime numbers • Proof: • Assume there are not infinitely many prime numbers, therefore they are listable, i.e. p1,p2,…,pn • Consider the number q = p1p2…pn+1. q is not divisible by any of the listed primes • Therefore, q is a prime. However, it was not listed. • Contradiction! Therefore, there are infinitely many primes.
Proof Techniques • Proof of Biimplications • To prove a theorem of the form ∀x (P(x) ↔ Q(x )), where D is the domain of the discourse, consider an arbitrary but fixed element a from D. For this a, prove that the biimplication P(a) ↔ Q(a) is true • The biimplication P↔ Q is equivalent to (P→ Q) ∧ (Q → P) • Prove that the implications P→ Q and Q → Pare true • Assume that Pis true and show that Q is true • Assume that Q is true and show that Pis true
Proof Techniques • Proof of Equivalent Statements • Consider the theorem that says that statements P,Q and r are equivalent • Show that P → Q, Q → R and R → P • Assume P and prove Q. Then assume Q and prove R Finally, assume R and prove P • What other methods are possible?
Other Proof Techniques • Vacuous • Trivial • Contrapositive • Counter Example • Divide into Cases • Constructive
Proof Basics You can not prove by example
Proof Strategies with Quantifiers • Existential • Constructive • some mathematicians only accept constructive proofs • Nonconstructive • show that denying existence leads to a contradiction • Universal • to prove false: • one counter-example • to prove true: • usually harder • the choose method
Proofs in Computer Science • Proof of program correctness • Proofs are used to verify approaches
Mathematical Induction • Assume that when a domino is knocked over, the next domino is knocked over by it • Show that if the first domino is knocked over, then all the dominoes will be knocked over
Mathematical Induction • Let P(n) denote the statement that then nth domino is knocked over • Base Step: Show that P(1) is true • Inductive Hypothesis: Assume some P(i) is true, i.e. the ith domino is knocked over for some • Inductive Step: Prove that P(i+1) is true, i.e.
Outline • Introduction • Sets • Logic & Boolean Algebra • Proof Techniques • Counting Principles • Combinatorics • Relations,Functions • Graphs/Trees • Boolean Functions, Circuits
Learning Objectives • Learn the basic counting principles—multiplication and addition • Explore the pigeonhole principle • Learn about permutations • Learn about combinations
Pigeonhole Principle • The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle.
