discrete computational structures

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


Functions


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.


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


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


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


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


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


Functions


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


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


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.


Functions


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.


Functions


Special Functions and Cardinality of a Set


Binary Operations

CSE 2353 OUTLINE

1. Sets 2. Logic

3. Proof Techniques



6. Functions

7. Counting Principles8. Boolean Algebra


Learning Objectives

Learn the basic counting principles—multiplication and addition

Explore the pigeonhole principle

Learn about permutations

Learn about combinations


Basic Counting Principles


Pigeonhole Principle

The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle.


Permutations


Combinations


Generalized Permutations and Combinations

CSE 2353 OUTLINE

1. Sets 2. Logic

3. Proof Techniques



6. Functions

7. Counting Principles

8. 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


Two-Element Boolean AlgebraLet B = {0, 1}.


Two-Element Boolean Algebra


Boolean Algebra


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 computational structures

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