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Introduction

This course provides an introduction to sets, set operations, mappings, relations, logic, formal languages, and computational complexity. It covers topics such as finite automata, graph theory, and analysis of algorithms.

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Introduction

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  1. Introduction • Tomáš Vaníček • Czech Technical University • Faculty of Civil Engeneering, Thákurova 7, Praha Dejvice, B407 • vanicek@fsv.cvut.cz • On CZU office 414, old building PEF

  2. Lecture Constents • Sets, set operations, mappings, cardinality • Relations and Operations • Logic, language PROLOG • Formal languages • Finite Automata, regular languages • Other formal model of computing • Computational complexity of algorithms • Graphs • Analysis of algorithms: graph, sortinh, searching of extrems, heuristic algorithms

  3. Literature • Vaníček,J., Papík,M., Pergl,R., Vaníček,T.: Teoretické základy informatiky, Kernberg Publishing, 2007 • Vaníček,J., Papík,M., Pergl,R., Vaníček,T.: Mathematical Foundation of Computer Science, Kernberg Publishing, 2008

  4. Set Theory • Set is a well defined collection of items • The question wheather element belongs to the set or no, must be clearly answered • Element x belongs to set A, x  A. • The element can be member of the set or no. • The element cannot be member of the set more times

  5. Classes and Sets • Sets can be members of aother sets • Russels paradox • Mathematic needs to deal with proper defined terms • For proper foundation of mathematic the axiomatic set theory is necessary

  6. Equality of sets, Subsets • Two sets are equal, if they have same members • Set with no elements is called empty: ø • Set A is subset of set B, if each element of A is element of B. • Set A if proper subset of B, if A is subset of B and A is not equal to B

  7. Describing of sets • By enumeration of elemnts • A={1,2,3} • B={Prague, Vienna, Budapest, Bartislava} • C={1,2,1,2,3,4,1}={1,2,3,4}={1,3,4,2} • By distinctive predicate A={x|P(x)} • A={x|x N, x<4} • B={x|x is capital of central European country} • By using already defined set • C={x N| x<4}

  8. Common set notations • Z… Set of Integers • Z+ … Set of positive integers • N… Set of nonnegative integers (natural numbers) • Q… Set of rational numbers • R… Set of real numbers • C… Set of complex numbers • R+, R-, Q+,…

  9. Set operations • Union • Intersection • Difference • Symmetric difference • Potency set

  10. Ordered pair • Ordered pair (a,b) is a set {{a,b},a}. a is assigned to be the first element. • Ordered n-tuple (a1,a2,…,an) could be defined by induction • For n=2 it is a ordered pair (a1,a2) • For n>2 it is a ordered pair of (n-1)-tuple (a2,…,an) and an element a1.

  11. Cartesian product • Cartesian product A x B is a set of all ordered pairs (a,b), where a is from A and b is from B. • Cartesian product of finite system of sets A1xA2x…xAn is a set of all n-tuples (a1,…,an) where ai is from Ai.

  12. Mappings • Mapping from set A to set B: for some elements of A there exist exactly one element of B • Mapping of set A to set B: for all elements of A there exist exactly one element of B • Mapping fromset A onto set B (surjection): Each element of B has its element in a such that m(a)=b (a is the pattern of b).

  13. Mappings • Simple mapping (injection): for different patterns a1,a2 there are different images b1,b2. • Bijection (one-to-one mapping) is injection A onto B.

  14. Cardinality of sets • Two finite sets A,B have same cardinality if there exist one-to-one mapping A onto B. • Cardinality of set A is represented as card(A), |A|, moh(A) • If card(A)≤card(B) there exist injection of A into B. • If card(A)≥card(B) there exist surjection of A onto B.

  15. Cardinality of infinite sets • Two infinite sets A,B have same cardinality if there exist one-to-one mapping A onto B. • card(N) = card(Z) = card(Q) = aleph0 • Set of all (infinite) lists of 0,1 (L) has bigger cardinality then aleph0. • card(L)=card(R). • card(2M)>card(M)

  16. Fuzziness • Reasons for uncertainty: • Stochastic character of the process (tomorrow will come rain) • Quantum uncertainty (temperature of water in the sink is 10 degrees) • Fuzziness of the term (I am a tall man)

  17. Fuzzy sets • Classical set theory: element belongs to set or no. • There exist characteristic function of set A, MA. • MA = 1, while x  A, MA = 0, while not x  A. • Fuzzy set is determined by characteristic function μA from universe U to interval <0,1> • μA (x)= 1, while x is definitely in A. • μA (x)= 0, while x is definitely not in A. • μA is between 0 and 1, while x is not definitely in A not definitely outside of A.

  18. Fuzzy sets • Support of A: supp(A)={xU|μA (x) > 0}. • Core of A: supp(A)={xU|μA (x) = 1}. • Height of fuzzy set: sup(supp(A)). • Normal fuzzy set: Height equal to 1. • α-level of the fuzzy set A {xU|μA (x) ≥ α}. • Α-cut of the fuzzy set A {xU|μA (x) = α}.

  19. Fuzzy set operations • A is subset of B: μA (x) ≤ μB(x) • B is difference of A: μB(x) = 1 - μA(x) • C is (standard) union of A and B: μC(x)=max(μA(x), μB(x)) • C is (standard) intersection of A and B: μC(x)=min(μA(x), μB(x))

  20. Fuzzy numbers • Let a≤b≤c≤d be four real numbers such that: • μA(x)=0 , for x<a and x>d • μA(x)=1 , for x between b and c • μA(x) is increasing between a and b. • μA(x)is decreasing between c and d. • Such set A we call fuzzy interval. • If b=c we call such a set fuzzy number.

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