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The equals sign, used to represent equality symbolically in an equation. In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object.
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality , which is concerned with whether the internal definitions of objects are the same.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
On Padé approximations to the exponential function and A-stable methods for the numerical solution of initial value problems (PDF) (Thesis). Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0.
Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
A proportion is a mathematical statement expressing equality of two ratios. [1] [2]: =: a and d are called extremes, b and c are called means.. Proportion can be written as =, where ratios are expressed as fractions.
It is an example of a decidable theory and is a fragment of more expressive decidable theories, including monadic class of first-order logic (which also admits unary predicates and is, via Skolem normal form, related [2] to set constraints in program analysis) and monadic second-order theory of a pure set (which additionally permits ...
The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final, axiom is a second-order statement of the principle of mathematical induction over the natural numbers, which makes this formulation close to second-order arithmetic .