What is Discrete Mathematics?
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451
Discrete Mathematics is a relative newcomer to the mathematics courses that undergraduates are encouraged or required to take. The origin of Discrete Mathematics courses involves the emergence of computer science as a major for undergraduate students. Computer science majors typically required a variety of mathematics skills to be successful practitioners of computer science. Discrete Mathematics courses, taught within Mathematics Departments, initially served this need. However, it soon became clear that the body of material that was being taught to computer science majors was also very very valuable for mathematics majors. This material offered a parallel entry track into mathematics that complemented but was different from the traditional entry track of Calculus courses.
In addition to the value of discrete mathematical ideas as a set of tools for mathematicians and computer scientists, discrete mathematics also lies at the core of a wide variety of techniques that have led to very important new technologies. Examples include Cell Phones, High Definition Television, DVD and CD technology, and the Internet.
Oversimplifying greatly, discrete mathematics is concerned with finite and discretely infinite collections collections of things. An example of a discretely infinite collection is the set of integers. There is no integer "between" 4 and 5 while there are infinitely many real numbers, even rational numbers between .000001 and .0000001! The major areas of mathematics which have come to be taught within the framework of a discrete mathematics course include sets, logic, number systems, relations and order relations, basic number theory, graph theory (especially trees), digraphs, counting (permutations and combinations), functions, proof techniques, finite-state machines and languages, and finite groups.
As a simple example of the way applications, computer science and mathematics interact with ideas in the domain of discrete mathematics, consider the following situation. The message "Send more money now" is to be sent using electronic mail. To carry this process out, the string of symbols in the English language must be converted to a string in computer readable form. Typically this means converting the string to some some sequence of binary symbols, 0's and 1's, because these are the building blocks for use by computer hardware. The binary string might be converted to "code" to protect the security of the message (think of a transaction with your bank about withdrawing money and paying a bill). This code, if subject to transmission error, could have important negative consequences, so one would want to protect the transmission using the concept of "error correction technology." The idea is to use a slightly longer string than the original string of binary digits so that when the new message string is received, if there are any errors that this can be detected and the errors corrected. Furthermore, there are ways of taking long files of 0's and 1's (arising from text, photographs, or TV images) and compressing these images so that they can be transmitted more quickly but reconstructed perfectly when they arrive. By using cryptography, error-correction, and data compression many of the modern technologies we have come accustomed to (wireless communication, DVD technology, etc.) are possible. Discrete mathematics stands behind all of these systems.