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CS 490

CS 490. Mathematical Logic, Combinatorics, Counting Arguments, Graph Theory, Number Theory, Discrete Probability, Recurrence Relation. Thao Tran. Mathematical Logic. Proposition and Logical Operators: Proposition:

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CS 490

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  1. CS 490 Mathematical Logic, Combinatorics, Counting Arguments, Graph Theory, Number Theory, Discrete Probability, Recurrence Relation. Thao Tran.

  2. Mathematical Logic • Proposition and Logical Operators: • Proposition: • A proposition is a sentence to which one and only one of the terms true o false can be meaningful applied. • Example: “four is even,” “43 >= 21” • Logical Operators: • Conjunction (And): If p and q are propositions, their conjunction, p and q (denoted p ^ q), is defined by: p q p ^ q 0 0 0 0 1 0 1 0 0 1 1 1 • Disjunction (Or): If p and q are propositions, their disjunction, p and q (denoted p v q), is defined by: p q p v q 0 0 0 0 1 1 1 0 1 1 1 1

  3. Mathematical Logic (cont.) • Negation: • if p is a proposition, its negation, not p, is denoted ~p and is defined by p ~p 0 1 1 0 • Conditional Operator( if…then): • The conditional statement if p then q, denoted p --> q , is defined by p q p  q 0 0 1 0 1 1 1 0 0 1 1 1 • Example: • If I pass the final, then I’ll graduate.

  4. Mathematical Logic( cont.) • Truth table:

  5. Mathematical Logic (cont.) • Tautology, Contradiction and Equivalent: • Tautology: An expression involving logical variables that is true in all cases of its truth table. • Example: p v ~p • Contradiction: An expression involving logical variable that is false in all cases of its truth table. • Example: p ^ ~p • Equivalent: Let S be a set of propositions and let r and s be propositions generated by S. r and s are equivalent if r <--> s is a tautology, denoted rs. • Example: p v q  q v p

  6. Counting Principles • It is frequently necessary to count how many ways certain choices can be made. • Basic methods: • Sum and product rules • Counting functions and sequences • Binomial theorem

  7. Sum and Product Rules • Rule of sum: • The number of ways in which either of two mutually exclusive events can occur is equal to the sum of the number of ways in which each can occur separately. • Rule of product: • The number of ways in which two independent events E1 and E2 can occur is the product of the numbers of ways in which E1 and E2 can occur separately.

  8. Sum and Product (cont.) • Example: • Suppose that a system of car registrartion is adopted in which and allowable registration plate consists of 1,2, or 3 letters, followed by a number (not starting with 0) having the same number of digits as there are letters. How many possible registration are there?

  9. Sum and Product (cont.) • Solution: • (26 x 9) + (26 x 26 x 9 x 10) + (26 x 26 x 26 x 9 x 10 x 10) = 15879474

  10. Binomial Theorems • Binomial Theorem • (x+y)n = ∑nr=0( nr) xryn-r • Where : ( nr) =n!/(n-r)!r!

  11. Power Set • Definition: • If A is any set, the power set of A is the set of all subsets of A, including the empty set and A itself. It is denoted P(A). • Example: • If A = {1, 2} then • P(A) = { φ, {1}, {2}, {1,2}} • Formula: • P(A) = 2#A

  12. Number Theory • The natural numbers: • N = {0, 1, 2, 3, …} • The integers: • Z = {…, -2, -1, 0, 1, 2, …} • Z stands for Zahlen, meaning “numbers” in Geman • The rational numbers: • Denoted Q (quotient), comprises all those numbers that can be written in the form a/b, with a,b in Z • The real numbers: • Denoted R: • Example: √2, π • The complex numbers: • The set C of complex numbers is the set of all numbers of the form a+bi where a and b are real numbers and i2 = -1 • The reason for extending from R to C is to be able to solve all polynomial equation

  13. Recurrence Relations • Definition: • Let S be a sequence of numbers. A recurrence relation on S is a formula that relates all but a finite number of terms of S to previous terms of S. That is, there is a k0 in the domain of S such that if k > k0, then S(k) is expressed in terms that preceed S(k). If the domain of S is {0, 1, 2…}, the terms S(0), S(1),…,S(k0) are not defined by the recurrence formula. Their values are the initial conditions (or boundary conditions, or basis) that complete the definition of S. • Example: • The Fibonacci sequence: • Fk = Fk-2 + Fk-1 , k >= 2 , F0=1, F1= 1 • This recurrence relation is called a second-order relation because Fk depends on the two previous terms of F.

  14. Recurrence Relations (cont.) • Solving a recurrence relation: • Sequence are often most easily defined with a recurrence relation; however, the calculation of terms by directly applying a recurrence relation can be time consuming. • Example: • Find recurrence relation for the sequence defined by: • D(k) = 5*2k , k>=0

  15. Recurrence Relations (cont.) • Answer: D(k) = 5*2k = 2*5*2k-1 = 2D(k-1) The relation is: D(k) – 2D(k-1) = 0 Initial condition D(0) = 5. • Homogeneous recurrent relation: • An nth order linear relation is a homogeneous recurrence relation if f(k) = 0 for all k. For each recurrence relation • S(k) + C1S(k-1)+…+CnS(k-n)=f(k) • The associated homogeneous relation is • S(k)+C1S(k-1)+ … + CnS(k-n)=0

  16. Graph Theory • Directed Graph: • Consist of a set of vertices, V, and a set of edges, E, connecting certain elements of V. Each element of E is an order pair. The first entry is the initial vertex of the edge and the second entry is the terminal vertex. • Example: • Simple Graph & Multigraph: • Simple graph is one for which there is no more than one edge directed from any one vertex to any other vertex. All other graphs are called multigraph.

  17. Graph Theory(cont.) • Traversals: • Eulerian Graph: • Konigsberg Bridge Problem:

  18. Graph Theory (cont.) • Answer: • A Eulerian path through a graph is a path whose edge list contains each edge of the graph exactly once.

  19. Graph Theory (cont.) • Hamiltonian Graph: • A hamiltonian path through a graph is a path whose vertex list contains each vertex of the graph exactly once.A hamiltonian graph is a graph that possesses a Hamiltonian path. • Traveling salesman problem : • A salesman who wants to minimize the number of miles the he travels in visiting his custommers.

  20. Discrete Probability • The calculation of discrete probability usually involves counting arguments • Permutation: • Prn , is called the number of permutations of n objects taken r at a time. • Prn = n(n-1)…(n-r+1) = n! / (n-r)! • Combination: • Define for an r-element subset of an n-element set A is a combination of A, taken r at a time. • Cnr = n! / r!(n-r)! • Example: Compute the number of distinct 5 card hands which can be dealt from a deck of 52 cards.

  21. Discrete Probability (cont.) • Answer: • C552 = 52!/(5! 47!) = 2,598,960 • The pigeonhole principle: • If n pigeons are assigned to m pigeonholes, and m < n, then at least one pigeonhole contains two or more pigeons. More generally, if n>km, then at least one pigeonhole must contain more than k pogeons.

  22. References • Truss, J.K. Discrete Mathematics for Computer Scientist. Addison-Wesley Publishing Company, 1991. • Kolman, Bernard and Robert. Discrete Mathematical Structures forComputer Science. Drexel University. • Doerr, Alan and Kenneth. Applied Discrete Structures for Computer Science. Science Research Associates, Inc. 1985.

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