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For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1. The empty relation R (defined so that aRb is never true) on a set X is vacuously symmetric and transitive; however, it is not reflexive (unless X itself is empty).
Trichlorofluoromethane, also called freon-11, CFC-11, or R-11, is a chlorofluorocarbon (CFC). It is a colorless, faintly ethereal, and sweetish-smelling liquid that boils around room temperature. [5] CFC-11 is a Class 1 ozone-depleting substance which damages Earth's protective stratospheric ozone layer. [6]
11 languages . বাংলা ... the same signature σ are called elementarily equivalent if they satisfy the same ... of Mathematics (3rd ed.), Elsevier, ISBN 978 ...
A relation R is both left and right Euclidean, if, and only if, the domain and the range set of R agree, and R is an equivalence relation on that set. [8] A right Euclidean relation is always quasitransitive, [9] as is a left Euclidean relation. [10] A connected right Euclidean relation is always transitive; [11] and so is a connected left ...
A torus knot is trivial iff either p or q is equal to 1 or −1. [4] [5] Each nontrivial torus knot is prime [6] and chiral. [4] The (p,q) torus knot is equivalent to the (q,p) torus knot. [3] [5] This can be proved by moving the strands on the surface of the torus. [7] The (p,−q) torus knot is the obverse (mirror image) of the (p,q) torus ...
Two rings R and S (associative, with 1) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over R, R-Mod, and the category of (left) modules over S, S-Mod. It can be shown that the left module categories R-Mod and S-Mod are equivalent if and only if the right module categories Mod-R and Mod-S are
Two metrics and on X are strongly or bilipschitz equivalent or uniformly equivalent if and only if there exist positive constants and such that, for every ,, (,) (,) (,).In contrast to the sufficient condition for topological equivalence listed above, strong equivalence requires that there is a single set of constants that holds for every pair of points in , rather than potentially different ...
[1] There is a similar notion of column equivalence, defined by elementary column operations; two matrices are column equivalent if and only if their transpose matrices are row equivalent. Two rectangular matrices that can be converted into one another allowing both elementary row and column operations are called simply equivalent.