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In this equality, A and B are distinguished by calling them left-hand side (LHS), and right-hand side (RHS). [3] Two objects that are not equal are said to be distinct. [4] Equality is often considered a kind of primitive notion, meaning, it is not formally defined, but rather informally said to be "a relation each thing bears to itself and ...
Sphere packing theorems in dimensions 8 and 24 (geometry, modular forms) Sphere theorem (Riemannian geometry) Spin–statistics theorem ; Sprague–Grundy theorem (combinatorial game theory) Squeeze theorem (mathematical analysis) Stahl's theorem (matrix analysis) Stallings theorem about ends of groups (group theory)
The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation. The rules can be expressed in English as: not (A or B) = (not A) and (not B) not (A and B) = (not A) or (not B) where "A or B" is an "inclusive or" meaning at least one of A or B rather than an "exclusive or" that means exactly one of A or B.
In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry. [4] Rauch comparison theorem relates the sectional curvature of a Riemannian manifold to the rate at which its geodesics spread apart; Toponogov's theorem; Myers's theorem; Hessian ...
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.
Examples of structures with two operations that are each distributive over the other are Boolean algebras such as the algebra of sets or the switching algebra. Multiplying sums can be put into words as follows: When a sum is multiplied by a sum, multiply each summand of a sum with each summand of the other sum (keeping track of signs) then add ...
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 ...
Consequently, a game lasts for at least 2n moves, and the number of pharisees is divisible by 4. A game of sprouts with n initial spots that ends in 2n moves. This lower bound on the length of a game is actually the minimum. The diagram on the right shows a completed game of 2n moves. It has n survivors, 2n neighbors and 0 pharisees.