Discrete Structures. Lecture 1
- 1. Discrete Structures (CS 335) Mohsin Raza University Institute of Information Technology PMAS Arid Agriculture University Rawalpindi
- 2. Discrete vs Continuous • Examples of discrete Data – Number of boys in the class. – Number of candies in a packet. – Number of suitcases lost by an airline. Discrete Continuous • Examples of continuous Data – Height of a person. – Time in a race. – Distance traveled by a car. 10/7/2013 Discrete Structures(CS 335) 2
- 3. What is discrete Structures? • Discrete mathematics is the part of mathematics devoted to the study of discrete objects (Kenneth H. Rosen, 6th edition). • Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous (wikipedia). 10/7/2013 Discrete Structures(CS 335) 3
- 4. Syllabus (Topics to be covered in this course) • • • • • • • • • • 10/7/2013 Logic Elementary Number Theory and Methods of Proof Set Theory Relations Sequences and Recursion Mathematical Induction Counting Relations and Equivalence Relations Graphs Trees Discrete Structures(CS 335) 4
- 5. Reference Books • Discrete Mathematics and its Applications (with Combinatorics and Graph Theory) 5th Edition, The McGraw-Hill Companies, 2007, Kenneth H. Rosen. • Discrete Mathematics with Applications 4th Edition, Thomson Learning, 1995, Susanna S. Epp. • Discrete Mathematics for Computer Scientists 2nd Edition, Addison-Wesley, 1999, John Truss. 10/7/2013 Discrete Structures(CS 335) 5
- 6. Logic • Propositional Logic • Logic of Compound Statements • Propositional Equivalences • Conditional Statements • Logical Equivalences • Valid and Invalid Arguments • Applications: Digital Logic Circuits • Predicates and Quantifiers • Logic of Quantified Statements 10/7/2013 Discrete Structures(CS 335) 6
- 7. Propositional Logic Proposition: A proposition (or Statement) is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both. Examples 1. Is the following sentence a proposition? If it is a proposition, determine whether it is true or false. Islamabad is the capital of Pakistan. This makes a declarative statement, and hence is a proposition. The proposition is TRUE (T). 10/7/2013 Discrete Structures(CS 335) 7
- 8. Examples (Propositions Cont.) 2. Is the following sentence a proposition? If it is a proposition, determine whether it is true or false. Can Ali come with you?. This is a question not the declarative sentence and hence not a proposition. 10/7/2013 Discrete Structures(CS 335) 8
- 9. Examples (Propositions Cont.) 3. Is the following sentence a proposition? If it is a proposition, determine whether it is true or false. Take two aspirins. This is an imperative sentence not the declarative sentence and therefore not a proposition. 10/7/2013 Discrete Structures(CS 335) 9
- 10. Examples (Propositions Cont.) 4. Is the following sentence a proposition? If it is a proposition, determine whether it is true or false. x+ 4 > 9. Because this is true for certain values of x (such as x = 6) and false for other values of x (such as x = 5), it is not a proposition. 10/7/2013 Discrete Structures(CS 335) 10
- 11. Examples (Propositions Cont.) 5. Is the following sentence a proposition? If it is a proposition, determine whether it is true or false. He is a college student. Because truth or falsity of this proposition depend on the reference for the pronoun he. it is not a proposition. 10/7/2013 Discrete Structures(CS 335) 11
- 12. Notations • The small letters are commonly used to denote the propositional variables, that is, variables that represent propositions, such as, p, q, r, s, …. • The truth value of a proposition is true, denoted by T or 1, if it is a true proposition and false, denoted by F or 0, if it is a false proposition. 10/7/2013 Discrete Structures(CS 335) 12
- 13. Compound Propositions Producing new propositions from existing propositions. Logical Operators or Connectives 1. Not 2. And ˄ 3. Or ˅ 4. Exclusive or 5. Implication 6. Biconditional 10/7/2013 Discrete Structures(CS 335) 13
- 14. Compound Propositions Negation of a proposition Let p be a proposition. The negation of p, denoted by p (also denoted by ~p), is the statement “It is not the case that p”. The proposition p is read as “not p”. The truth values of the negation of p, p, is the opposite of the truth value of p. 10/7/2013 Discrete Structures(CS 335) 14
- 15. Examples 1. Find the negation of the following proposition p : Today is Friday. The negation is p : It is not the case that today is Friday. This negation can be more simply expressed by p : Today is not Friday. 10/7/2013 Discrete Structures(CS 335) 15
- 16. Examples 2. Write the negation of “6 is negative”. The negation is “It is not the case that 6 is negative”. or 10/7/2013 “6 is nonnegative”. Discrete Structures(CS 335) 16
- 17. Truth Table (NOT) • Unary Operator, Symbol: p true false false 10/7/2013 p true Discrete Structures(CS 335) 17
- 18. Conjunction (AND) Definition Let p and q be propositions. The conjunction of p and q, denoted by p˄q, is the proposition “p and q”. The conjunction p˄q is true when p and q are both true and is false otherwise. 10/7/2013 Discrete Structures(CS 335) 18
- 19. Examples 1. Find the conjunction of the propositions p and q, where p : Today is Friday. q : It is raining today. The conjunction is p˄q : Today is Friday and it is raining today. 10/7/2013 Discrete Structures(CS 335) 19
- 20. Truth Table (AND) • Binary Operator, Symbol: p pq true true true true false false false true false false 10/7/2013 q false false Discrete Structures(CS 335) 20
- 21. Disjunction (OR) Definition Let p and q be propositions. The disjunction of p and q, denoted by p˅q, is the proposition “p or q”. The disjunction p˅q is false when both p and q are false and is true otherwise. 10/7/2013 Discrete Structures(CS 335) 21
- 22. Examples 1. Find the disjunction of the propositions p and q, where p : Today is Friday. q : It is raining today. The disjunction is p˅q : Today is Friday or it is raining today. 10/7/2013 Discrete Structures(CS 335) 22
- 23. Truth Table (OR) • Binary Operator, Symbol: p pq true true true true false true false true true false 10/7/2013 q false false Discrete Structures(CS 335) 23
- 24. Exclusive OR (XOR) Definition Let p and q be propositions. The exclusive or of p and q, denoted by pq, is the proposition “pq”. The exclusive or, p q, is true when exactly one of p and q is true and is false otherwise. 10/7/2013 Discrete Structures(CS 335) 24
- 25. Examples 1. Find the exclusive or of the propositions p and q, where p : Atif will pass the course CSC102. q : Atif will fail the course CSC102. The exclusive or is pq : Atif will pass or fail the course CSC102. 10/7/2013 Discrete Structures(CS 335) 25
- 26. Truth Table (XOR) • Binary Operator, Symbol: p pq true true false true false true false true true false 10/7/2013 q false false Discrete Structures(CS 335) 26
- 27. Examples (OR vs XOR) The following proposition uses the (English) connective “or”. Determine from the context whether “or” is intended to be used in the inclusive or exclusive sense. 1. “Nabeel has one or two brothers”. A person cannot have both one and two brothers. Therefore, “or” is used in the exclusive sense. 10/7/2013 Discrete Structures(CS 335) 27
- 28. Examples (OR vs XOR) 2. To register for BSC you must have passed the qualifying exam or be listed as an Math major. Presumably, if you have passed the qualifying exam and are also listed as an Math major, you can still register for BCS. Therefore, “or” is inclusive. 10/7/2013 Discrete Structures(CS 335) 28
- 29. Composite Statements Statements and operators can be combined in any way to form new statements. p q p q (p)(q) true true false false false true false false true true false true true false true false false true true true 10/7/2013 Discrete Structures(CS 335) 29
- 30. Lecture Summery • Introduction to the Course • Propositions • Logical Connectives • Truth Tables • Compound propositions 10/7/2013 Discrete Structures(CS 335) 30