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The concept of proof is formalized in the field of mathematical logic. [ 12] A formal proof is written in a formal language instead of natural language. A formal proof is a sequence of formulas in a formal language, starting with an assumption, and with each subsequent formula a logical consequence of the preceding ones.
Objects studied in discrete mathematics include integers, graphs, and statements in logic. [ 1][ 2][ 3] By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been ...
Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. [ 1 ...
Description. The simplest and most common form of mathematical induction infers that a statement involving a natural number n (that is, an integer n ≥ 0 or 1) holds for all values of n. The proof consists of two steps: The base case (or initial case ): prove that the statement holds for 0, or 1. The induction step (or inductive step, or step ...
Automated theorem proving. Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major motivating factor for the development of computer science .
Propositional calculus. The propositional calculus[ a] is a branch of logic. [ 1] It is also called propositional logic, [ 2] statement logic, [ 1] sentential calculus, [ 3] sentential logic, [ 1] or sometimes zeroth-order logic. [ 4][ 5] It deals with propositions [ 1] (which can be true or false) [ 6] and relations between propositions, [ 7 ...
Stars and bars (combinatorics) In the context of combinatorial mathematics, stars and bars (also called "sticks and stones", [ 1] "balls and bars", [ 2] and "dots and dividers" [ 3]) is a graphical aid for deriving certain combinatorial theorems. It can be used to solve many simple counting problems, such as how many ways there are to put n ...
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. [ 1] This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning .
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