discrete computational structures
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DISCRETE COMPUTATIONAL STRUCTURES
CSE 2353
Spring 2006
Final Slides
CSE 2353 OUTLINE
1. Sets 2. Logic
3. Proof Techniques
4. Integers and Induction
5. Relations and Posets
6. Functions7. Counting Principles
8. Boolean Algebra
Discrete Mathematical Structures: Theory and Applications 3
Learning Objectives
Learn about functions
Explore various properties of functions
Learn about binary operations
Discrete Mathematical Structures: Theory and Applications 4
Functions
Discrete Mathematical Structures: Theory and Applications 5
Discrete Mathematical Structures: Theory and Applications 6
Discrete Mathematical Structures: Theory and Applications 7
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 8
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:
1) 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
2) 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 9
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 10
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 11
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 12
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 13
Functions
Discrete Mathematical Structures: Theory and Applications 14
Functions
Discrete Mathematical Structures: Theory and Applications 15
Functions 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.
Example 5.1.16
Discrete Mathematical Structures: Theory and Applications 16
Functions
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.
Example 5.1.18
Discrete Mathematical Structures: Theory and Applications 17
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 18
Functions
Discrete Mathematical Structures: Theory and Applications 19
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 20
Functions
Discrete Mathematical Structures: Theory and Applications 21
Functions
Discrete Mathematical Structures: Theory and Applications 22
Special Functions and Cardinality of a Set
Discrete Mathematical Structures: Theory and Applications 23
Special Functions and Cardinality of a Set
Discrete Mathematical Structures: Theory and Applications 24
Special Functions and Cardinality of a Set
Discrete Mathematical Structures: Theory and Applications 25
Special Functions and Cardinality of a Set
Discrete Mathematical Structures: Theory and Applications 26
Special Functions and Cardinality of a Set
Discrete Mathematical Structures: Theory and Applications 27
Special Functions and Cardinality of a Set
Discrete Mathematical Structures: Theory and Applications 28
Special Functions and Cardinality of a Set
Discrete Mathematical Structures: Theory and Applications 29
Special Functions and Cardinality of a Set
Discrete Mathematical Structures: Theory and Applications 30
Discrete Mathematical Structures: Theory and Applications 31
Special Functions and Cardinality of a Set
Discrete Mathematical Structures: Theory and Applications 32
Special Functions and Cardinality of a Set
Discrete Mathematical Structures: Theory and Applications 33
Special Functions and Cardinality of a Set
Discrete Mathematical Structures: Theory and Applications 34
Special Functions and Cardinality of a Set
Discrete Mathematical Structures: Theory and Applications 35
Discrete Mathematical Structures: Theory and Applications 36
Special Functions and Cardinality of a Set
Discrete Mathematical Structures: Theory and Applications 37
Binary Operations
Discrete Mathematical Structures: Theory and Applications 38
Discrete Mathematical Structures: Theory and Applications 39
Discrete Mathematical Structures: Theory and Applications 40
CSE 2353 OUTLINE
1. Sets 2. Logic
3. Proof Techniques
4. Integers and Induction
5. Relations and Posets
6. Functions
7. Counting Principles8. Boolean Algebra
Discrete Mathematical Structures: Theory and Applications 42
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 43
Basic Counting Principles
Discrete Mathematical Structures: Theory and Applications 44
Basic Counting Principles
Discrete Mathematical Structures: Theory and Applications 45
Basic Counting Principles
Discrete Mathematical Structures: Theory and Applications 46
Pigeonhole Principle
The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle.
Discrete Mathematical Structures: Theory and Applications 47
Pigeonhole Principle
Discrete Mathematical Structures: Theory and Applications 48
Discrete Mathematical Structures: Theory and Applications 49
Pigeonhole Principle
Discrete Mathematical Structures: Theory and Applications 50
Permutations
Discrete Mathematical Structures: Theory and Applications 51
Permutations
Discrete Mathematical Structures: Theory and Applications 52
Combinations
Discrete Mathematical Structures: Theory and Applications 53
Combinations
Discrete Mathematical Structures: Theory and Applications 54
Generalized Permutations and Combinations
Discrete Mathematical Structures: Theory and Applications 55
Generalized Permutations and Combinations
CSE 2353 OUTLINE
1. Sets 2. Logic
3. Proof Techniques
4. Integers and Induction
5. Relations and Posets
6. Functions
7. Counting Principles
8. Boolean Algebra
Discrete Mathematical Structures: Theory and Applications 57
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 58
Two-Element Boolean AlgebraLet B = {0, 1}.
Discrete Mathematical Structures: Theory and Applications 59
Two-Element Boolean Algebra
Discrete Mathematical Structures: Theory and Applications 60
Discrete Mathematical Structures: Theory and Applications 61
Discrete Mathematical Structures: Theory and Applications 62
Discrete Mathematical Structures: Theory and Applications 63
Two-Element Boolean Algebra
Discrete Mathematical Structures: Theory and Applications 64
Two-Element Boolean Algebra
Discrete Mathematical Structures: Theory and Applications 65
Discrete Mathematical Structures: Theory and Applications 66
Discrete Mathematical Structures: Theory and Applications 67
Discrete Mathematical Structures: Theory and Applications 68
Discrete Mathematical Structures: Theory and Applications 69
Discrete Mathematical Structures: Theory and Applications 70
Boolean Algebra
Discrete Mathematical Structures: Theory and Applications 71
Boolean Algebra
Discrete Mathematical Structures: Theory and Applications 72
Logical Gates and Combinatorial Circuits
Discrete Mathematical Structures: Theory and Applications 73
Logical Gates and Combinatorial Circuits
Discrete Mathematical Structures: Theory and Applications 74
Logical Gates and Combinatorial Circuits
Discrete Mathematical Structures: Theory and Applications 75
Logical Gates and Combinatorial Circuits
Discrete Mathematical Structures: Theory and Applications 76
Discrete Mathematical Structures: Theory and Applications 77
Discrete Mathematical Structures: Theory and Applications 78
Discrete Mathematical Structures: Theory and Applications 79
Discrete Mathematical Structures: Theory and Applications 80
Discrete Mathematical Structures: Theory and Applications 81
Discrete Mathematical Structures: Theory and Applications 82
Discrete Mathematical Structures: Theory and Applications 83
Discrete Mathematical Structures: Theory and Applications 84
Discrete Mathematical Structures: Theory and Applications 85
Discrete Mathematical Structures: Theory and Applications 86
Discrete Mathematical Structures: Theory and Applications 87
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 88
Discrete Mathematical Structures: Theory and Applications 89
Discrete Mathematical Structures: Theory and Applications 90
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