Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.[1] The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems
In mathematics, axiomatization is the formulation of a system of statements (i.e. axioms) that relate a number of primitive terms in order that a consistent body of propositions may be derived deductively from these statements. Thereafter, the proof of any proposition should be, in principle, traceable back to these axioms
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects
Giuseppe Peano (Italian: [dʒuˈzɛppe peˈaːno]; 27 August 1858 – 20 April 1932) was an Italian mathematician. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction. He spent most of his career teaching mathematics at the University of Turin
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.[7][8] Mathematicians seek out patterns[9][10] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution)
Arithmetic or arithmetics (from the Greek ἀριθμός arithmos, "number") is the oldest[1] and most elementary branch of mathematics. It consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are sometimes still used to refer to a wider part of number theory.[2
The harmonica, also known as a French harp or mouth organ,[1] is a free reed wind instrument used worldwide in many musical genres, notably in blues, American folk music, classical music, jazz, country, and rock and roll. There are many types of harmonica, including diatonic, chromatic, tremolo, octave, orchestral, and bass versions. A harmonica is played by using the mouth (lips and tongue) to direct air into or out of one or more holes along a mouthpiece. Behind each hole is a chamber containing at least one reed. A harmonica reed is a flat elongated spring typically made of brass, stainless steel, or bronze, which is secured at one end over a slot that serves as an airway. When the free end is made to vibrate by the player's air, it alternately blocks and unblocks the airway to produce sound
A wind instrument is a musical instrument that contains some type of resonator (usually a tube), in which a column of air is set into vibration by the player blowing into (or over) a mouthpiece set at the end of the resonator. The pitch of the vibration is determined by the length of the tube and by manual modifications of the effective length of the vibrating column of air. In the case of some wind instruments, sound is produced by blowing through a reed; others require buzzing into a metal mouthpiece
Reed aerophones is one of the categories of musical instruments found in the Hornbostel-Sachs system of musical instrument classification. In order to produce sound with these Aerophones the player's breath is directed against a lamella or pair of lamellae which periodically interrupt the airflow and cause the air to be set in motion. 422 Reed aerophones Hne 422.1 Double reed instruments - There are two lamellae which beat against one another. 422.11 (Single) oboes. 422.111 With cylindrical bore. 422.111.1 Without fingerholes. 422.111.2 With fingerholes. Duduk Piri 422.112 With conical bore. Oboe Oboe d'amore Bassoon Taepyeongso 422.12 Sets of oboes. 422.121 With cylindrical bore. 422.122 With conical bore.
Woodwind instruments are a family of musical instruments within the more general category of wind instruments. There are two main types of woodwind instruments: flutes and reed instruments (otherwise called reed pipes). What differentiates these instruments from other wind instruments is the way in which they produce their sound.[1] Examples are a saxophone, a bassoon and a piccolo