Outline of discrete mathematics

From Wikipedia, the free encyclopedia

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[1] – do not vary smoothly in this way, but have distinct, separated values.[2] Discrete mathematics, therefore, excludes topics in "continuous mathematics" such as calculus and analysis.

Included below are many of the standard terms used routinely in university-level courses and in research papers. This is not, however, intended as a complete list of mathematical terms; just a selection of typical terms of art that may be encountered.

Discrete mathematical disciplines[edit]

For further reading in discrete mathematics, beyond a basic level, see these pages. Many of these disciplines are closely related to computer science.

Concepts in discrete mathematics[edit]

Sets[edit]

Functions[edit]

  • Function – Association of one output to each input
  • Domain of a function – Mathematical concept
  • Codomain – Target set of a mathematical function
  • Range of a function – Subset of a function's codomain
  • Image (mathematics) – Set of all values of a function
  • Injective function – Function that preserves distinctness
  • Surjection – Function such that every element has a preimage (mathematics)
  • Bijection – One-to-one correspondence
  • Function composition – Operation on mathematical functions
  • Partial function – Function whose actual domain of definition may be smaller than its apparent domain
  • Multivalued function – Generalized mathematical function
  • Binary function – Function that takes two inputs
  • Floor function – Nearest integers from a number
  • Sign function – Mathematical function returning -1, 0 or 1
  • Inclusion map – or inclusion function, or canonical injection
  • Pigeonhole principle – If there are more items than boxes holding them, one box must contain at least two items
  • Relation composition – Mathematical operation
  • Permutations – Mathematical version of an order change
  • Symmetry – Mathematical invariance under transformations

Arithmetic[edit]

Elementary algebra[edit]

Elementary algebra – Basic concepts of algebra

Mathematical relations[edit]

Mathematical phraseology[edit]

  • If and only if – Logical connective
  • Necessary and sufficient – Terms to describe a conditional relationship between two statements
  • Distinct – Relationship asserting that two quantities are the same
  • Difference – One of the four basic arithmetic operations
  • Absolute value – Distance from zero to a number
  • Up to – Mathematical statement of uniqueness, except for an equivalent structure (equivalence relation)
  • Modular arithmetic – Computation modulo a fixed integer
  • Characterization (mathematics) – Term in mathematics
  • Normal form – Standard representation of a mathematical object
  • Canonical form – Standard representation of a mathematical object
  • Without loss of generality – Expression in mathematics
  • Vacuous truth – Conditional statement which is true because the antecedent cannot be satisfied
  • Contradiction – Logical incompatibility between two or more propositions, Reductio ad absurdum – Argument that leads to a logical absurdity
  • Counterexample – Exception to a proposed general rule
  • Sufficiently large – mathematical concept
  • Pons asinorum – Statement that the angles opposite the equal sides of an isosceles triangle are themselves equal
  • Table of mathematical symbols – Meanings of symbols used in mathematics
  • Contrapositive – Mathematical logic concept
  • Mathematical induction – Form of mathematical proof

Combinatorics[edit]

Combinatorics – Branch of discrete mathematics

Probability[edit]

Probability – Branch of mathematics concerning chance and uncertainty

  • Average – Number taken as representative of a list of numbers
  • Expected value – Average value of a random variable
  • Discrete random variable – Variable representing a random phenomenon
  • Sample space – Set of all possible outcomes or results of a statistical trial or experiment
  • Event – In statistics and probability theory, set of outcomes to which a probability is assigned
  • Conditional Probability – Probability of an event occurring, given that another event has already occurred
  • Independence – When the occurrence of one event does not affect the likelihood of another
  • Random variables – Variable representing a random phenomenon

Propositional logic[edit]

  • Logical operator – Symbol connecting sentential formulas in logic
  • Truth table – Mathematical table used in logic
  • De Morgan's laws – Pair of logical equivalences
  • Open sentence – formula that contains at least one free variable
  • List of topics in logic – Overview of and topical guide to logic

Mathematicians associated with discrete mathematics[edit]

  • Paul Erdős – Hungarian mathematician (1913–1996)
  • Leonhard Euler - Swiss mathematician (1707-1783)
  • Richard Karp - American computer scientist (b. 1935)
  • Ronald Graham – American mathematician (1935–2020)
  • Claude Shannon - American mathematician (1916-2001)
  • Donald Knuth - American mathematician and computer scientist (b. 1938)
  • George Szekeres – Hungarian-Australian mathematician
  • Aristotle – Ancient Greek philosopher and polymath (384–322 BC)

See also[edit]

References[edit]

  1. ^ Richard Johnsonbaugh, Discrete Mathematics, Prentice Hall, 2008; James Franklin, Discrete and continuous: a fundamental dichotomy in mathematics, Journal of Humanistic Mathematics 7 (2017), 355-378.
  2. ^ Weisstein, Eric W. "Discrete mathematics". MathWorld.

External links[edit]