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Unformatted text preview: MAT067 University of California, Davis Winter 2007 Some Common Mathematical Symbols and Abbreviations (with History) Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (January 21, 2007) Binary Relations = (the equals sign ) means “is the same as” and was first introduced in the 1557 book The Whetstone of Witte by Robert Recorde (c. 1510-1558). He wrote, ”I will sette as I doe often in woorke use, a paire of parralles, or Gemowe lines of one lengthe, thus : ==, bicause noe 2, thynges, can be moare equalle.” (Recorde used an elongated form of the modern equals sign.) < (the less than sign ) mean “is strictly less than”, and > (the greater than sign ) means “is strictly greater than”. They first appeared in Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas (“The Analytical Arts Applied to Solving Algebraic Equations”) by Thomas Harriot (1560-1621), which was published posthumously in 1631. Pierre Bouguer (1698-1758) later refined these to ≤ (“less than or equals”) and ≥ (“greater than or equals”) in 1734. := (the equal by definition sign ) means “is equal by definition to”. This is a common alternate form of the symbol “= Def ”, which appears in the 1894 book Logica Matematica by the logician Cesare Burali-Forti (1861–1931). Other common alternate forms of the symbol “= Def ” include “ def =” and “ ≡ ”, the latter being especially common in applied mathematics. Some Symbols from Mathematical Logic ∴ ( three dots ) means “therefore” and first appeared in print in the 1659 book Teusche Algebra (“Teach Yourself Algebra”) by Johann Rahn (1622-1676). 3 (the such that sign) means “under the condition that”. However, it is much more common (and less ambiguous) to just abbreviate “such that” as “s.t.”. ⇒ (the implies sign) means “logically implies that”. (E.g., “if it’s raining, then it’s pouring” is equivalent to saying “it’s raining ⇒ it’s pouring.”) The history of this symbol is unclear. ⇐⇒ (the iff sign) means “if and only if” and is used to connect logically equivalent statements. (E.g., “it’s raining iff it’s really humid” means simultaneously that “if it’s raining, then it’s Copyright c 2007 by the authors. These lecture notes may be reproduced in their entirety for non-commercial purposes. really humid” and that “if it’s really humid, then it’s raining”. In other words, the statement “it’s raining” implies the statement “it’s really humid” and vice versa.) This notation “iff” is attributed to the great mathematician Paul R. Halmos (1916–2006)....
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