The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. Besides reading the book, students are strongly encouraged to do all the excercises. Mathematics is a discipline in which working the problems is essential to the understanding of the material contained in this book.
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Solutions to problems found at the end of this book can only be used when you are stuck. Exert a reasonable amount of efforts towards solving a problem before you look up the answer, and rework any problem you miss. Discrete Mathematics: An Open Introduction is a free, open source textbook appropriate for a first or second year undergraduate course for math majors, especially those who will go on to teach. Since Spring , the book has been used as the primary textbook or a supplemental resource at multiple universities around the world see the partial adoptions list.
This 3rd edition brings many improvements, including nearly new exercises, a new section on trees in the graph theory chapter, and improved exposition throughout. Previous editions will continue to be available indefinitely.
Please contact the author with feedback and suggestions, or if you are decide to use the book in a course you are teaching. The entire book is available for free as an interactive online ebook. This should work well on all screen sizes, including smart phones. Hints and solutions to examples and exercises are hidden but easily revealed by clicking on their links.
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Some exercises also allow you to enter and check your work, so you can try multiple times without spoiling the answer. For offline use, a free pdf version , suitable for reading on a tablet or computer, is available for download. This should be searchable and easy to navigate using embedded links. Hints and solutions when available can be accessed by clicking on the exercise number, and clicking on the number of the hint or solution will bring you back to the exercise. If you prefer a physical copy, an inexpensive print version of the text is available on Amazon. How- ever, there is no requirement that a function be given by a formula.
The modern concept of function has been obtained from the formula-based idea by abstracting basic properties the old concept had and using them as the basis of the new definition. This process of definition by abstracting properties is a major tool in mathematics, and you will see more examples of it later in the book see Chapter 51, for example. The concept of function as a formula never disappeared entirely, but was studied mostly by logicians who generalized it to the study of function-as-algorithm. This is an oversimplification of history.
Of course, the study of algorithms is one of the central topics of modern computer science, so the notion of function-as-formula more generally, function-as-algorithm has achieved a new importance in recent years. Nevertheless, computer science needs the abstract definition of function given here. Functions such as sin may be and quite often are programmed to look up their values in a table instead of calculating them by a formula, an arrangement which gains speed at the expense of using more memory. The graph of a function Cartesian product 52 coordinate 49 F is constructed this way: FC.
Some important types of functions tion 61 identity function 63 identity 72 Observe that the graph of inc is the same as the graph of idA and they have the same domain, so that the only difference between them is what is considered the codomain A for idA , B for the inclusion of A in B.
A constant function from R to R has a horizontal line as its graph. Such a function is an empty function. Its graph is empty, and it has no values. An empty function has nothing to do. Anonymous notation for functions The curly-brackets notation for sets has the advantage that it allows you to refer to a set without giving it a name. A notation which describes without naming is called anonymous notation.
The notation we have introduced for functions does not have that advantage. When the two versions of the squaring function were discussed, it was necessary to call them S and T in order to say anything about them. The older one is called lambda notation and is used mostly in logic and computer science.
The straight arrow goes from domain to codomain, the barred arrow from definition 4 element of the domain to element of the codomain. A and B are any sets. Predicates determine functions What can you say about P? Sets of functions As mathematical entities, functions can be elements of sets; in fact the discovery of function spaces, in which functions are regarded as points in a space, was one of the great advances of mathematics.
It is easy to read this notation backward. For each item in the first column, state binary operation 67 which of the items in the second column it is an element of. Binary operations powerset 46 real number 12 right band 67 P is called the right band on A. Fixes prefix notation 68 reverse Polish nota- tion 68 This is called prefix notation for functions and is familiar from analytic geometry and calculus texts.
Pascal and many other computer languages require the parentheses, however, and Mathematica requires square brackets. It is important not to confuse evaluation written like this with multiplication. This is called infix notation and naturally applies only to functions with two arguments. This is called postfix notation. This has real advantages which will become apparent when we look at composition in Chapter Note the use of spaces to separate the items.
Writing functions on the right which are normally infixed, without parentheses, is naturally called reverse Polish nota- tion. The language Lisp uses prefix notation with parentheses and the various Cartesian product 52 dialects of Forth characteristically use reverse Polish notation no parentheses. Fix notation in Mathematica Mathematica gives the user control over whether a function is written in infix notation or not. For example, we remarked in Sec- tion More about binary operations Associativity The opera- tion given in These are both associative binary operations on the set R of real numbers.
Give Law 71 reasons for your answer. Commutativity We will not even 5 prove that law here. Identities powerset 46 right band 67 These are two different ideas, but there is a relationship between them see What is the identity element for the binary operation of union on PS? What is the identity element for the binary operation of intersection on PS? Cartesian product 52 commutative 71 Proof This follows immediately from Definition Logic texts often define a relation to be a predicate of this type, but the point of view taken here that a relation is a set of ordered pairs is most common in mathematics and computer science.
Here is the ordered pair 49 relation in Example Digraphs are studied in depth in Chapters and Relations on a single set Cartesian product 52 codomain 56 Relations and functions However, a functional relation in Rel A, B is always the graph of a function whose domain is some subset of A. For them, the domain and codomain are not part of the definition.
In general, a relation ordered pair 49 powerset 46 associates a possibly empty subset of B to each element of A. Operations on relations definition 4 equivalence 40 Reflexive relations The choice of 0.
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The relations S and N will be used several times below in examples. Reflexivity and irreflexivity are properties of the relation and the set it is defined on, not of particular elements of the set on which the relation is defined.
This comment also applies to the other properties of relations discussed in this section. Symmetric relations The sister relation S Example Thus a it could be vacuously true the empty relation is symmetric!
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Antisymmetric relations Orderings are covered in Chapter Another problem like this is Problem Transitive relations That equals is transitive is equivalent to the statement from high-school geometry that two things equal to the same thing are equal to each other. In fact, all the properties so far have been defined by implications except reflexivity. And indeed the empty relation is symmetric, antisymmetric and transi- tive!
HINT: There are 14 possible com- binations and two impossible ones.
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Irreflexive relations If you divide n into m you get a quotient and a remainder; for example, if you divide 4 into 14 you get a quotient 3 and a remainder 2.