 Discrete mathematics. Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying smoothly the objects studied in discrete mathematics, such as integers, graphs and statements in logic, do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in continuous mathematics such as calculus or Euclidean geometry. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets, finite sets or sets with the same cardinality as the natural numbers. However, there is no exact definition of the term discrete mathematics. Indeed, discrete mathematics is described less by what is included than by what is excluded, continuously varying quantities and related notions. The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes the applied two parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business. Research in discrete mathematics increased in the latter half of the 20th century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research. Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well. In university curricula, discrete mathematics appeared in the 1980s, initially as a computer science support course, that's contents were somewhat haphazard at the time. The curriculum has thereafter developed in conjunction with efforts by ACM and MA into a course that is basically intended to develop mathematical maturity in freshmen, therefore it is nowadays the prerequisite for mathematics majors in some universities as well. Some high school level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics is sometimes seen as a preparatory course, not unlike pre-calculus in this respect. The Fulcrum Prize is awarded for outstanding papers in discrete mathematics.