Abstract
We introduce and discuss here the fundamental concept of function.
We also recall the notion of set and of some general mathematical notation.
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Notes
- 1.
The symbol & is often used in logic in place of ∧. Logicians more often write the implication symbol ⇒ as → and the relation of logical equivalence as ←→ or ↔. However, we shall adhere to the symbolism indicated in the text so as not to overburden the symbol →, which has been traditionally used in mathematics to denote passage to the limit.
- 2.
G.W. Leibniz (1646–1716) – outstanding German scholar, philosopher, and mathematician to whom belongs the honor, along with Newton, of having discovered the foundations of the infinitesimal calculus.
- 3.
H. Poincaré (1854–1912) – French mathematician whose brilliant mind transformed many areas of mathematics and achieved fundamental applications of it in mathematical physics.
- 4.
Galileo Galilei (1564–1642) – Italian scholar and outstanding scientific experimenter. His works lie at the foundation of the subsequent physical concepts of space and time. He is the father of modern physical science.
- 5.
The notation \(A\Rightarrow B\Rightarrow C\) will be used as an abbreviation for \((A\Rightarrow B)\wedge(B\Rightarrow C)\).
- 6.
Bourbaki, N. “The architecture of mathematics” in: N. Bourbaki, Elements of the history of mathematics, translated from the French by John Meldrum, Springer, New York, 1994.
- 7.
G. Cantor (1845–1918) – German mathematician, the creator of the theory of infinite sets and the progenitor of set-theoretic language in mathematics.
- 8.
B. Russell (1872–1970) – British logician, philosopher, sociologist and social activist.
- 9.
A. de Morgan (1806–1871) – Scottish mathematician.
- 10.
R. Descartes (1596–1650) – outstanding French philosopher, mathematician and physicist who made fundamental contributions to scientific thought and knowledge.
- 11.
P. Fermat (1601–1665) – remarkable French mathematician, a lawyer by profession. He was one of the founders of a number of areas of modern mathematics: analysis, analytic geometry, probability theory, and number theory.
- 12.
H.A. Lorentz (1853–1928) – outstanding Dutch theoretical physicist. Poincaré called these transformations Lorentz transformations in honor of Lorentz, who stimulated the research of symmetries in Maxwell’s equations. They were used by Einstein in 1905 in the formulation of his theory of special relativity.
- 13.
Johann Bernoulli (1667–1748) – one of the early representatives of the distinguished Bernoulli family of Swiss scholars; he studied analysis, geometry and mechanics. He was one of the founders of the calculus of variations. He gave the first systematic exposition of the differential and integral calculus.
- 14.
S.F. Lacroix (1765–1843) – French mathematician and educator (professor at the École Normale and the École Polytechnique, and member of the Paris Academy of Sciences).
- 15.
N.I. Lobachevskii (1792–1856) – great Russian scholar, to whom belongs the credit – shared with the great German scientist C.F. Gauss (1777–1855) and the outstanding Hungarian mathematician J. Bólyai (1802–1860) – for having discovered the non-Euclidean geometry that bears his name.
- 16.
Lobachevskii, N.I. Complete Works, Vol. 5, Moscow–Leningrad: Gostekhizdat, 1951, p. 44 (Russian).
- 17.
For the sake of completeness it is useful to note that a relation ℛ is reflexive if its domain of definition and its range of values are the same and the relation \(a\mathcal{R}a\) holds for any element \(a\) in the domain of ℛ.
- 18.
R. Dedekind (1831–1916) – German algebraist who took an active part in the development of the theory of a real number. He was the first to propose the axiomatization of the set of natural numbers usually called the Peano axiom system after G. Peano (1858–1932), the Italian mathematician who formulated it somewhat later.
- 19.
F. Bernstein (1878–1956) – German mathematician, a student of G. Cantor. E. Schröder (1841–1902) – German mathematician.
- 20.
J. von Neumann (1903–1957) – American mathematician who worked in functional analysis, the mathematical foundations of quantum mechanics, topological groups, game theory, and mathematical logic. He was one of the leaders in the creation of the first computers.
- 21.
E. Zermelo (1871–1953) – German mathematician. A. Fraenkel (1891–1965) – German (later, Israeli) mathematician.
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Zorich, V.A. (2015). Some General Mathematical Concepts and Notation. In: Mathematical Analysis I. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48792-1_1
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