𝗣𝗗𝗙 | These are notes on discrete mathematics for computer scientists. Graphs are mathematical structures that have many applications in computer science. Using Discrete Mathematics in Computer Science 87 .. increasing use of discrete mathematical structures to clarify and explain key concepts and . changes we make will be posted at medical-site.info pdf. Doing some basic discrete mathematics. ▷ Getting a Chapters 1 and 8 of Mathematics for Computer Science by . Web link: medical-site.info∼lekheng/ flt/medical-site.info Also the multiplicative structure (N, 1,·) of natural numbers with one.
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Notes on Discrete Mathematics So why do I need to learn all this nasty mathematics? 1. But isn't . Functions on recursive structures. single gigantic PDF file at medical-site.info In many computer science departments, discrete mathematics is one of the first courses data structures course is not a prerequisite for this course. We feel that . Mathematical Structures for Computer Science: A Modern Treatment of Discrete Mathematics Discrete Mathematical Structures. Read more.
This is all done through the discrete math concept of sets. Sets allow information to be grouped and put in order. Since each piece of information and each trait belonging to that piece of information is discrete, the organization of such information in a database requires discrete mathematical methods. Uses for Discrete Math in Logistics Logistics is the study of organizing the flow of information, goods and services. Without discrete mathematics, logistics would not exist.
This is because logistics makes heavy use of graphs and graph theory, a sub-field of discrete math. Graph theory allows complex logistical problems to simplify into graphs consisting of nodes and lines. A mathematician can analyze these graphs according to the methods of graph theory to determine the best routes for shipping or solving other logistical problems.
Computer Algorithms Algorithms are the rules by which a computer operates. These rules are created through the laws of discrete mathematics. A computer programmer uses discrete math to design efficient algorithms.
This design includes applying discrete math to determine the number of steps an algorithm needs to complete, which implies the speed of the algorithm. The third provides a way to count the number of elements in a collection of sets in which some of the sets may contain some of the same elements that the other sets contain. The fourth and fifth deal with important proof techniques called the Principleof MathematicalInduction and the Strong Form of MathematicalInduction.
We use induction to find the set of elements for which a statement about the integers is true. An important application of the Principle of Mathematical Induction, in both its forms, is to show how algorithms can be proven to be correct without any execution by a computer. Basic Definitions The idea of a set is simple: A set is a collection of elements.
The " What is the basic characteristic of a set? For any set A and any element b, either b is in A or b is not in A. If you ask whether an element is in a set, the answer is either yes or no. The expression "is an element of" is denoted by the symbol E, a form of the Greek letter epsilon.
Is a member of, is contained in, or simply is in means the same as "is an element of. Despite its frequent use, the term set is not defined in terms of other concepts. Like the terms point and line in plane geometry, set is a primitive concept. Just assume there are elements, there are sets, and that for a set A and an element b, the assertion b E A is either true or false. They are not the same. The number of times an element is listed and the order in which elements are listed are both unimportant.
Consequently, these two sets contain the same elements. That is, these two sets are equal, as we shall see later. The first is by a list of all the elements. The second is by a description of some property the elements have.
The third is by a description based on some other sets. The "language" used to describe sets is called settheoretic notation.
Nobody said the pattern had to be simple. This notation should only be used when it is obvious from the context exactly what is meant.
The most famous is Russell's paradox, which is similar to the so-called "liar's paradox": "This sentence is a lie. Russell's paradox is this. Let x be the set of all sets that are not elements of themselves. Now, is x an element of itself? Work through it: If it is, then it is not, and if it is not, then it is.
What's wrong? Most modem set theorists assert that using definition method 2 is at fault-note that Bertrand Russell English mathematician and philosopher, used that form in defining x.
The set of all sets which are not elements of themselves is deemed "too big" to be a set. By using definition method 3, we avoid constructing sets which are "too big. For example, 1, 2, 22, 2', We call such a set a sequence. We can refer to a sequence by ao, al, a The notion of a sequence will be examined more carefully in Section 4. At this time, we just need to have a way to refer to sets of this form.
Special Sets There are special names for certain common sets of numbers. Some of them are listed here. For example, we can define the notion of a number being a prime without knowing that any particular number is a prime. We must then show that any number we think is a prime has the defining property.
First, we need to know what a divisor is before we can define a prime. A natural number p is prime if p : 1 and its only divisors are 1 and p. In Example 1, we will show that P is nonempty. Example 1. Prove that 3 is a prime-that is, that 3 E P. We must show 3 has the property that its only divisors are 1 and 3. Since the only other possibility is 2 and 2 does not divide 3, 3 is therefore a prime.
U A divisor of an integer is also called a factor. For example, what does it mean for two sets to be equal? Basic Definitions 5 Definition 1.