In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
typically used to establish that a given statement is true of all natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s (positive integers). It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.
The method can be extended to prove statements about more general well-founded structures, such as trees
Tree (set theory)
In set theory, a tree is a partially ordered set In set theory, a tree is a partially ordered set (poset) In set theory, a tree is a partially ordered set (poset) (T, In set theory, a tree is a partially ordered set (poset) (T, ...
; this generalization, known as structural induction
Structural induction is a proof method that is used in mathematical logic , computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction...
, is used in mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
and computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...