DM Lecture 6.pdf - DISCRETE STRUCTURES COURSE I NSTRUCTOR MUHAMMAD SAIF UL I S LAM Course Outline Logic and Proofs(Chapter 1 Sets and Functions(Chapter

# DM Lecture 6.pdf - DISCRETE STRUCTURES COURSE I NSTRUCTOR...

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DISCRETE STRUCTURES COURSE INSTRUCTOR: MUHAMMAD SAIF UL ISLAM

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Course Outline Logic and Proofs (Chapter 1) Sets and Functions (Chapter 2) Relations Number Theory © Saif ul Islam - 2017 Combinatorics and Recurrence Graphs Trees Discrete Probability FAST NUCES – DISCRETE STRUCTURES 2
Lecture Outline Definition of a Function. o Domain, Cdomain o Image, Preimage Injection, Surjection, Bijection © Saif ul Islam - 2017 Inverse Function Function Composition Graphing Functions Floor, Ceiling, Factorial Partial Functions (optional) FAST NUCES – DISCRETE STRUCTURES 3

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Functions Definition : Let A and B be nonempty sets. A function f from A to B , denoted f : A B is an assignment of each element of A to exactly one element of B . We write f ( a ) = b if b is the unique element of B assigned by the function f to the element a of A . Functions are sometimes called mappings or transformations . © Saif ul Islam - 2017 FAST NUCES – DISCRETE STRUCTURES 4
Functions Let G be the function that assigns a grade to a student in our discrete mathematics class. The domain of G is the set X= {Adams, Chou, Goodfriend, Rodriguez, Stevens} The codomain of G is the set Y= { A, B, C, D, F } The range of G is the set { A, B,C, F } © Saif ul Islam - 2017 X Y G (Adams) = A G (Stevens) = F FAST NUCES – DISCRETE STRUCTURES 5

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Functions Given a function f : A B : We say f maps A to B or f is a mapping from A to B . A is called the domain of f . B is called the co-domain of f . © Saif ul Islam - 2017 If f ( a ) = b , then b is called the image of a under f . a is called the pre-image of b. The range of f is the set of all images of points in A under f . We denote it by f ( A ). Two functions are equal when they have the same domain, the same co-domain and map each element of the domain to the same element of the co-domain. FAST NUCES – DISCRETE STRUCTURES 6
Functions © Saif ul Islam - 2017 FAST NUCES – DISCRETE STRUCTURES 7

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Functions © Saif ul Islam - 2017 FAST NUCES – DISCRETE STRUCTURES 8
Representing Functions Functions may be specified in different ways: An explicit statement of the assignment. o Students and grades example. A formula. © Saif ul Islam - 2017 o f ( x ) = x + 1 A computer program. o A Java program that when given an integer n , produces the n th Fibonacci Number (covered in the next section and also in Chapter 5 ).

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