This document provides an overview of mathematical preliminaries including sets, functions, relations, graphs, and proof techniques. It defines sets, set operations, and set representations. It also covers functions, relations, Cartesian products, graphs including walks, paths, cycles, and trees. Finally, it discusses proof techniques like induction and contradiction and provides examples of proofs using these techniques.
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
CMSC 56 | Lecture 12: Recursive Definition & Algorithms, and Program Correctnessallyn joy calcaben
Recursive Definition & Algorithms, and Program Correctness
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 23, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
CMSC 56 | Lecture 12: Recursive Definition & Algorithms, and Program Correctnessallyn joy calcaben
Recursive Definition & Algorithms, and Program Correctness
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 23, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Date: March 9, 2016
Course: UiS DAT911 - Foundations of Computer Science (fall 2016)
Please cite, link to or credit this presentation when using it or part of it in your work.
Theory of automata and formal languageRabia Khalid
KleenE Star Closure, Plus operation, recursive definition of languages, INTEGER, EVEN, factorial, PALINDROME, languages of strings, cursive definition of RE, defining languages by RE,Examples
Date: March 9, 2016
Course: UiS DAT911 - Foundations of Computer Science (fall 2016)
Please cite, link to or credit this presentation when using it or part of it in your work.
Theory of automata and formal languageRabia Khalid
KleenE Star Closure, Plus operation, recursive definition of languages, INTEGER, EVEN, factorial, PALINDROME, languages of strings, cursive definition of RE, defining languages by RE,Examples
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Sets & Set Operation
CMSC 56 | Discrete Mathematical Structure for Computer Science
September 11, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
I am Gabriel C. I love exploring new topics. Academic writing seemed an interesting option for me. After working for many years with programmingexamhelp.com, I have assisted many students with their exams. I can proudly say, each student I have served is happy with the quality of the solution that I have provided. I have acquired Business analyst of Information Technology, Montreal College of Information Technology, Canada.
3. A set is a collection of elements SETS We write 1 is a member (or element) of set A ship is not a member (or element) of set B Membership of a given set
14. Powersets A powerset is a set of subsets Powerset of S = the set of all the subsets of S S = { a, b, c } 2 S = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} } Observation: | 2 S | = 2 |S| ( 8 = 2 3 )
15. Cartesian Product A = { 2, 4 } B = { 2, 3, 5 } A X B = { (2, 2), (2, 3), (2, 5), ( 4, 2), (4, 3), (4, 5) } A X B is an ordered set, i.e. A X B ≠ B X A |A X B| = |A|·|B| Generalizes to more than two sets A X B X … X Z
16.
17. FUNCTIONS domain 1 2 3 a b c range f : A -> B A B If A = domain then f is a total function otherwise f is a partial function f(1) = a 4 5 In general, we mean this.
20. Walk Walk is a sequence of adjacent edges (e, d), (d, c), (c, a) is a walk from e to a of length 3 (or denoted as e-d-c-a ) Length = # of edges a b c d e
21. Path a b c d e Path is a walk where no edge is repeated Simple path : no node is repeated
22. Path a b c d e Path is a walk where no edge is repeated Simple path : no node is repeated (e, b), (b, e), (e, d), (d, c), (c, a) is a path from e to a but it is not a simple path.
23. Cycle a b c d e 1 2 3 Cycle : a walk from a node (base) to itself without repeated edges Simple cycle : only the base node is repeated Loop: an edge from a node to itself base
24. Find All Simple Paths starting from c a b c d e origin The longest simple path has at most length 4. Since every vertex can only be visited at most once, and there are 4 other vertices.
25. (c, a) (c, e) Step 1 a b c d e origin Starting from vertex c, list all outgoing edges as long as they do not lead to any vertex already used in the path. At this point, we have all paths of length one starting at c . For all vertices a , e reached by c , we list all outgoing edges originating at a or e according the same way we did before.
26. (c, a) (c, a), (a, b) (c, e) (c, e), (e, b) (c, e), (e, d) Step 2 a b c d e origin
27. Step 3 a b c d e origin (c, a) (c, a), (a, b) (c, a), (a, b), (b, e) (c, e) (c, e), (e, b) (c, e), (e, d)
28. Step 4 a b c d e origin (c, a) (c, a), (a, b) (c, a), (a, b), (b, e) (c, a), (a, b), (b, e), (e,d) (c, e) (c, e), (e, b) (c, e), (e, d)
29. Trees are connected directed graphs without cycles such that there is a special vertex called “root” having exactly one path to every other vertices. root leaf parent child
30. root leaf Level 0 Level 1 Level 2 Level 3 Height 3 The level associated with each vertex is the number of edges in the path form the root to the vertex. The height of the tree is the largest level number of any vertex.
31. Binary Trees A binary tree is a tree in which no parent can have more than two children. A binary tree is a tree in which no parent can have more than two children. (p.10) Example 1.5. Prove that a binary tree of height n has at most 2 n leaves.
32.
33.
34.
35. Example Theorem: A binary tree of height n has at most 2 n leaves. (p.10) We want to show: L(n) ≦ 2 n for n = 0, 1, 2,…. Proof by induction: let L(i) be the maximum number of leaves of any subtree at height i
36.
37. Induction Step From Inductive hypothesis: height k k+1 Let’s assume L(i) ≦ 2 i for all i = 0, 1, …, k need to show that L(k + 1) ≦ 2 k+1 0 … L(k) ≦ 2 k
38. L(k) ≦ 2 k L(k+1) ≦ 2 · L(k) ≦ 2· 2 k = 2 k+1 Induction Step height k k+1 (we add at most two nodes for every leaf of level k) … need to show that L(k + 1) ≦ 2 k+1 To get a binary tree of height k+1 from one of height k , we can create at most 2 leaves in place of each previous one
39.
40.
41.
42. = n/m 2 m 2 = n 2 Therefore, n 2 is even n is even n = 2 k 2 m 2 = 4k 2 m 2 = 2k 2 m is even m = 2 p Thus, m and n have a common factor 2 Contradiction!