DISCRETE COMPUTATIONAL STRUCTURES

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Presentation transcript:

DISCRETE COMPUTATIONAL STRUCTURES CSE 2353 Spring 2006 Final Slides

Integers and Induction CSE 2353 OUTLINE Sets Logic Proof Techniques Integers and Induction Relations and Posets Functions Counting Principles Boolean Algebra

Learning Objectives Learn about functions Explore various properties of functions Learn about binary operations Discrete Mathematical Structures: Theory and Applications

Functions Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Functions Every function is a relation Therefore, functions on finite sets can be described by arrow diagrams. In the case of functions, the arrow diagram may be drawn slightly differently. If f : A → B is a function from a finite set A into a finite set B, then in the arrow diagram, the elements of A are enclosed in ellipses rather than individual boxes. Discrete Mathematical Structures: Theory and Applications

Functions To determine from its arrow diagram whether a relation f from a set A into a set B is a function, two things are checked: Check to see if there is an arrow from each element of A to an element of B This would ensure that the domain of f is the set A, i.e., D(f) = A Check to see that there is only one arrow from each element of A to an element of B This would ensure that f is well defined Discrete Mathematical Structures: Theory and Applications

Functions Let A = {1,2,3,4} and B = {a, b, c , d} be sets The arrow diagram in Figure 5.6 represents the relation f from A into B Every element of A has some image in B An element of A is related to only one element of B; i.e., for each a ∈ A there exists a unique element b ∈ B such that f (a) = b Discrete Mathematical Structures: Theory and Applications

Functions Therefore, f is a function from A into B The image of f is the set Im(f) = {a, b, d} There is an arrow originating from each element of A to an element of B D(f) = A There is only one arrow from each element of A to an element of B f is well defined Discrete Mathematical Structures: Theory and Applications

Functions The arrow diagram in Figure 5.7 represents the relation g from A into B Every element of A has some image in B D(g ) = A For each a ∈ A, there exists a unique element b ∈ B such that g(a) = b g is a function from A into B Discrete Mathematical Structures: Theory and Applications

Functions The image of g is Im(g) = {a, b, c , d} = B There is only one arrow from each element of A to an element of B g is well defined Discrete Mathematical Structures: Theory and Applications

Functions Discrete Mathematical Structures: Theory and Applications

Functions Discrete Mathematical Structures: Theory and Applications

Functions Example 5.1.16 Let A = {1,2,3,4} and B = {a, b, c , d}. Let f : A → B be a function such that the arrow diagram of f is as shown in Figure 5.10 The arrows from a distinct element of A go to a distinct element of B. That is, every element of B has at most one arrow coming to it. If a1, a2 ∈ A and a1 = a2, then f(a1) = f(a2). Hence, f is one-one. Each element of B has an arrow coming to it. That is, each element of B has a preimage. Im(f) = B. Hence, f is onto B. It also follows that f is a one-to-one correspondence. Discrete Mathematical Structures: Theory and Applications

Functions Let A = {1,2,3,4} and B = {a, b, c , d, e} Example 5.1.18 Let A = {1,2,3,4} and B = {a, b, c , d, e} f : 1 → a, 2 → a, 3 → a, 4 → a For this function the images of distinct elements of the domain are not distinct. For example 1  2, but f(1) = a = f(2) . Im(f) = {a}  B. Hence, f is neither one-one nor onto B. Discrete Mathematical Structures: Theory and Applications

Functions Let A = {1,2,3,4} and B = {a, b, c , d, e} f : 1 → a, 2 → b, 3 → d, 4 → e For this function, the images of distinct elements of the domain are distinct. Thus, f is one-one. In this function, for the element c of B, the codomain, there is no element x in the domain such that f(x) = c ; i.e., c has no preimage. Hence, f is not onto B. Discrete Mathematical Structures: Theory and Applications

Functions Discrete Mathematical Structures: Theory and Applications

Functions Let A = {1,2,3,4}, B = {a, b, c , d, e},and C = {7,8,9}. Consider the functions f : A → B, g : B → C as defined by the arrow diagrams in Figure 5.14. The arrow diagram in Figure 5.15 describes the function h = g ◦ f : A → C. Discrete Mathematical Structures: Theory and Applications

Functions Discrete Mathematical Structures: Theory and Applications

Functions Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

Binary Operations Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Integers and Induction CSE 2353 OUTLINE Sets Logic Proof Techniques Integers and Induction Relations and Posets Functions Counting Principles Boolean Algebra

Learning Objectives Learn the basic counting principles— multiplication and addition Explore the pigeonhole principle Learn about permutations Learn about combinations Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

Pigeonhole Principle The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle. Discrete Mathematical Structures: Theory and Applications

Pigeonhole Principle Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Pigeonhole Principle Discrete Mathematical Structures: Theory and Applications

Permutations Discrete Mathematical Structures: Theory and Applications

Permutations Discrete Mathematical Structures: Theory and Applications

Combinations Discrete Mathematical Structures: Theory and Applications

Combinations Discrete Mathematical Structures: Theory and Applications

Generalized Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

Generalized Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

Integers and Induction CSE 2353 OUTLINE Sets Logic Proof Techniques Integers and Induction Relations and Posets Functions Counting Principles Boolean Algebra

Learning Objectives Learn about Boolean expressions Become aware of the basic properties of Boolean algebra Explore the application of Boolean algebra in the design of electronic circuits Learn the application of Boolean algebra in switching circuits Discrete Mathematical Structures: Theory and Applications

Two-Element Boolean Algebra Let B = {0, 1}. Discrete Mathematical Structures: Theory and Applications

Two-Element Boolean Algebra Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Two-Element Boolean Algebra Discrete Mathematical Structures: Theory and Applications

Two-Element Boolean Algebra Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Boolean Algebra Discrete Mathematical Structures: Theory and Applications

Boolean Algebra Discrete Mathematical Structures: Theory and Applications

Logical Gates and Combinatorial Circuits Discrete Mathematical Structures: Theory and Applications

Logical Gates and Combinatorial Circuits Discrete Mathematical Structures: Theory and Applications

Logical Gates and Combinatorial Circuits Discrete Mathematical Structures: Theory and Applications

Logical Gates and Combinatorial Circuits Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Logical Gates and Combinatorial Circuits The Karnaugh map, or K-map for short, can be used to minimize a sum-of-product Boolean expression. Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications