This document provides an introduction and overview of a discrete mathematics course. It explains that discrete mathematics lays the mathematical foundations for many computer science topics. It lists common topics covered in the course like logic, sets, functions, counting, recursion, graphs and trees. It emphasizes that discrete mathematics deals with discrete, distinct objects like those used in computing. Mastering this subject involves actively practicing the concepts rather than just hearing or reading about them.

1. Why Are You Studying this Course
 This course will develop your mathematical maturity
2
 Discrete mathematics provides the mathematical
foundations for many computer science courses
• Data Structures
• Algorithm Analysis and Design
• Database Management Systems and Database Theory
• Compiler Construction
• Computer Security
• Digital Logic Design
• Artificial Intelligence

2. 3
Course Information
 Syllabus:
• Logic
• Sets
• Relations
• Functions.
• Combinatory: counting, permutations, combinations.
• Recursion
• Probibility
• Mathematical Induction
• Graph theory: Graphs and Trees

3. 4
Course Information
 Textbook:
• Discrete Mathematics and its Applications by Kenneth. H.
Rosen, 7th Edition
• Discrete Mathematics with Application” by Susana. 4th
edition, 2010

4. 5
 How will you master Discrete Structures
• “I hear and I forget. I see and I remember. I do and
understand” - Chinese proverb

5. 6
Terminology
Discrete - Composed of distinct OR separable/unconnected parts.
(Opposite of continuous.)
Structures - Objects built up from simpler objects according to
some definite pattern.

6. 7

7. 8
Introduction
 What is discrete mathematics?
• Part of mathematics devoted to the study of
discrete objects.
• Discrete means consisting of distinct or
unconnected elements.
• As computers are discrete object operating one
jumpy, discontinuous step at a time, Discrete
Math is the right framework for describing
precisely Computer Science concepts.

8. 9
Introduction
 What is discrete mathematics?
• In computer science we usually deal with finite,
discrete objects.
• For example, we cannot store a real number
(infinite precision) in a computer but can only
store bits (finite precisions).
 Definition
• Discrete Mathematics is a collection of
mathematical topics that examine and use finite
or countable infinite mathematical objects.

9. Problems Solved Using DM

10. Logic – 7

11. Statement – 8a

12. Examples – 8b

13. Truth Values of Propositions – 8c

14. Examples – 9a

15. Statements & Truth Values – 9b
T
T
F
F

16. Example – 10b

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19. Example – 11b

20. Compound Statement – 12a

21. Symbolic Representation – 13a

22. Logical Connectives – 14a

23. 24
Some Alternative Notations
Name: not and or xor implies iff
Propositional logic:      
Boolean algebra: p pq + 
C/C++/Java (wordwise): ! && || != ==
C/C++/Java (bitwise): ~ & | ^
Logic gates:

24. Examples – 14b

25. Translating from English to
Symbols – 15

26. Translating from English to
Symbols – 16a

27. Translating from English to
Symbols – 16

28. Translating from English to
Symbols – 17a

29. Translating from English to
Symbols – 17b

30. Negation – 19

31. Truth Table for ~p – 20

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33. Conjunction – 21

34. Truth Table for p ^ q – 22

35. Disjunction – 23

36. Truth Table for p q – 15

37. Truth Table

38. • Advanced versions of many Internet search engines allow you
to use some form of and, or and not to refine the search
process.
• Imagine that you want to find web pages about careers in
mathematics or computer science but not finance or
marketing.
• With a search engine that uses quotation marks to enclose
exact phrases and expresses and as AND, or as OR, and not as
NOT, you would write
• Careers AND (mathematics OR "computer science")
• AND NOT (finance OR marketing).

39. Truth Table for ~p^q - 2

40. Truth Table for ~p^q – 2a

41. Truth Table for ~p^q – 2b

42. Truth Table for ~p^q – 2c

43. ~p ^ (q v~ r) – (2 - 3a)

44. ~p ^ (q v~ r) – 2 - 3b

45. ~p ^ (q v~ r) – 2 - 3c

46. ~p ^ (q v~ r) – 2 - 3d

47. v
Truth Table for ~p (p v~ q) – 2 - 3e

48. Truth Table for (pvq) ^~ (p^q) – 2 - 4a

49. Truth Table for (pvq) ^~ (p^q) – 2 - 4c

50. v v
Truth Table for (pvq) ^~ (p^q) – 2 - 4e

51. Truth Table for (pvq) ^~ (p^q) – 2 -4f

52. Exclusive OR – 2 - 5

53. Symbols for Exclusive OR – 2 - 5a
p q p q
T
T
F
T
F
T
F
T
T
F
F
F

54. Bit Strings
Thus voltage memory stored in a computer can be
represented by a sequence of 0’s and 1’s such as
01 1011 0010 1001
Another portion of the memory might look like
10 0010 1111 1001
Each of the number in the sequence is called a bit, and
the whole sequence of bits is called a bit string.
Lecture 1 55

55. Logical Equivalence – 2 - 6

56. Double Negation ~(~p) ≡ p – 2 - 7

57. Examples – 2 - 12

58. Example – 2 - 17c

59. Example – 2 - 17d

60. Example – 2 - 17e

61. Exercise – 2 - 19

62. Tautology – 2 - 21

63. Example – 2 - 21a

64. Contradiction – 2 - 22

65. Example – 2 - 22a

66. Exercise – 2 - 23

67. Prove that the above statement is a Tautology
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68. Exercise – 2 - 24