DM Lecture 3.pdf - DISCRETE STRUCTUERS COURSE I NSTRUCTOR MUHAMMAD SAIF UL I SLAM Course Outline Logic and Proofs(Chapter 1 Sets and Functions Relations

DM Lecture 3.pdf - DISCRETE STRUCTUERS COURSE I NSTRUCTOR...

This preview shows page 1 - 9 out of 39 pages.

DISCRETE STRUCTUERS COURSE INSTRUCTOR: MUHAMMAD SAIF UL ISLAM
Image of page 1

Subscribe to view the full document.

© Saif ul Islam - 2017 Course Outline Logic and Proofs (Chapter 1) Sets and Functions Relations Number Theory Combinatorics and Recurrence Graphs Trees Discrete Probability FAST NUCES DISCRETE STRUCTURES 2
Image of page 2
© Saif ul Islam - 2017 Lecture Outline Limitations of Propositional Logic Predicates Quantifiers Logical Equivalences Involving Quantifiers Translating from English to Logic Examples Nested Quantifiers Order of Quantifiers Translating Nested Quantifiers FAST NUCES DISCRETE STRUCTURES 3
Image of page 3

Subscribe to view the full document.

© Saif ul Islam - 2017 Limitations of Propositional Logic Propositional logic cannot adequately express the meaning of all statements in mathematics and in natural language Example: If we want to say “ All students are present.” OR “ Some students are present ”, We need N number of statement to represent it. Where N refers to the number of students . Can’t be represented in propositional logic. Need a language that talks about objects, their properties, and their relations. FAST NUCES DISCRETE STRUCTURES 4
Image of page 4
© Saif ul Islam - 2017 Predicates Predicate logic can be used to express the meaning of a wide range of statements in mathematics and computer science in ways that permit us to reason and explore relationships between objects Predicate logic uses the following new features: Variables: x , y , z Predicates: P ( x ), M ( x ) Quantifiers ( to be covered in a few slides ): Propositional functions are a generalization of propositions. They contain variables and a predicate, e.g., P ( x ) Variables can be replaced by elements from their domain . FAST NUCES DISCRETE STRUCTURES 5
Image of page 5

Subscribe to view the full document.

© Saif ul Islam - 2017 Predicates Example: The statement x is greater than 3” has two parts. 1- The variable x , is the subject of the statement. 2- The predicate , “is greater than 3” refers to a property that the subject of the statement can have. We can denote the statement “ x is greater than 3” by P ( x ), where P denotes the predicate “is greater than 3” and x is the variable. FAST NUCES DISCRETE STRUCTURES 6
Image of page 6
© Saif ul Islam - 2017 Propositional Functions Once a value has been assigned to the variable x , the statement P ( x ) becomes a proposition and has a truth value Example: Let P ( x ) denote the statement “ x > 3 .” What are the truth values of P (4) and P (2)? We obtain the statement P (4) by setting x = 4 in the statement “ x > 3 .” P (4 ), which is the statement “4 > 3,” is true . P (2 ), which is the statement “2 > 3 ,” is false . FAST NUCES DISCRETE STRUCTURES 7
Image of page 7

Subscribe to view the full document.

© Saif ul Islam - 2017 Propositional Functions Example: Let Q ( x, y ) denote the statement “ x = y + 3 .” What are the truth values of the propositions Q (1 , 2) and Q (3 , 0)?
Image of page 8
Image of page 9
  • Fall '19

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes