Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. The term "discrete mathematics" is therefore used in contrast with "continuous mathematics," which is the branch of mathematics dealing with objects that can vary smoothly (and which includes, for example, calculus). Whereas discrete objects can often be characterized by integers, continuous objects require real numbers.

The study of how discrete objects combine with one another and the probabilities of various outcomes is known as combinatorics. Other fields of mathematics that are considered to be part of discrete mathematics include graph theory and the theory of computation. Topics in number theory such as congruences and recurrence relations are also considered part of discrete mathematics.

The study of topics in discrete mathematics usually includes the study of algorithms, their implementations, and efficiencies. Discrete mathematics is the mathematical language of computer science, and as such, its importance has increased dramatically in recent decades.

Portions of this entry contributed by John Renze

Balakrishnan, V. K. Introductory Discrete Mathematics. New York: Dover, 1997.

Bobrow, L. S. and Arbib, M. A. Discrete Mathematics: Applied Algebra for Computer and Information Science. Philadelphia, PA: Saunders, 1974.

Dossey, J. A.; Otto, A. D.; Spence, L.; and Eynden, C. V. Discrete Mathematics, 3rd ed. Reading, MA: Addison-Wesley, 1997.

Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.

Hall, C. and O'Donnell, J. Discrete Mathematics Using a Computer. London: Springer-Verlag, 2000.

Lipschutz, S. and Lipson, M. L. 2000 Solved Problems in Discrete Mathematics. New York: McGraw-Hill, 1991.

Lipschutz, S. and Lipson, M. L. Schaum's Outline of Discrete Mathematics, 2nd ed. New York: McGraw-Hill, 1997.

Rosen, K. Applications of Discrete Mathematics, 4th ed. New York: McGraw-Hill, p. 1998.

Rosenstein, J. G.; Franzblau, D. S.; and Roberts, F. S. Discrete Mathematics in the Schools. Providence, RI: Amer. Math. Soc., 1997.

Skiena, S. Implementing Discrete Mathematics. Reading, MA: Addison-Wesley, 1990.

Weisstein, E. W. "Books about Discrete Mathematics." http://www.ericweisstein.com/encyclopedias/books/DiscreteMathematics.html.

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, 2002.

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Renze, John and Weisstein, Eric W. "Discrete Mathematics." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiscreteMathematics.html