Browse Definitions :
Definition

mathematical induction

Mathematical induction is a a specialized form of deductive reasoning used to prove a fact about all the elements in an infinite set by performing a finite number of steps. 

In order for mathematical induction to work with an infinite set, that set must be denumerable, meaning that a one-to-one correspondence must exist between the elements of the set in question and the set of positive integers. In other words, it must be possible to express the set in the form an implied list of discrete elements such as {1, 2, 3, 4, ...}.

Consider a denumerably (also called countably) infinite set X with elements x1, x2, x3, x4, and so on. In order to prove a proposition about all the elements of X, we begin by proving that the proposition holds true for x1, the first element in the set X. Then we must prove that if the proposition holds true for some arbitrary element xn in X (where n is a positive integer), then the proposition also holds true for the next element xn+1 in set X. If we can do both of these two things successfully using deductive reasoning, we create an infinite chain of true statements by rigorous logical implication, proving the proposition true for all of the elements in X.

The first explicit formalization of the induction principle was composed by the French mathematician Blaise Pascal in 1665. Mathematical induction should not be confused with inductive reasoning. The former principle is mathematically rigorous (meaning that the conclusions are logically certain), but the latter methodology deals with probability and allows for some uncertainty.

This was last updated in June 2013

Start the conversation

Send me notifications when other members comment.

Please create a username to comment.

-ADS BY GOOGLE

File Extensions and File Formats

SearchCompliance

SearchSecurity

  • cybersecurity

    Cybersecurity is the protection of internet-connected systems, including hardware, software and data, from cyberattacks.

  • asymmetric cryptography (public key cryptography)

    Asymmetric cryptography, also called public key cryptography, uses a pair of numerical keys that are mathematically related to ...

  • digital signature

    A digital signature is a mathematical technique used to validate the authenticity and integrity of a message, software or digital...

SearchHealthIT

SearchDisasterRecovery

  • business continuity plan (BCP)

    A business continuity plan (BCP) is a document that consists of the critical information an organization needs to continue ...

  • disaster recovery team

    A disaster recovery team is a group of individuals focused on planning, implementing, maintaining, auditing and testing an ...

  • cloud insurance

    Cloud insurance is any type of financial or data protection obtained by a cloud service provider. 

SearchStorage

  • hard disk drive (HDD)

    A computer hard disk drive (HDD) is a non-volatile memory hardware device that controls the positioning, reading and writing of ...

  • byte

    In most computer systems, a byte is a unit of data that is eight binary digits long. Bytes are often used to represent a ...

  • network-attached storage (NAS)

    Network-attached storage (NAS) is dedicated file storage that enables multiple users and heterogeneous client devices to retrieve...

Close