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The examples "is greater than", "is at least as great as", and "is equal to" are transitive relations on various sets. As are the set of real numbers or the set of natural numbers: whenever x > y and y > z, then also x > z whenever x ≥ y and y ≥ z, then also x ≥ z whenever x = y and y = z, then also x = z. More examples of transitive ...
A transitive set (or class) that is a model of a formal system of set theory is called a transitive model of the system (provided that the element relation of the model is the restriction of the true element relation to the universe of the model). Transitivity is an important factor in determining the absoluteness of formulas.
Protein folding problem: Is it possible to predict the secondary, tertiary and quaternary structure of a polypeptide sequence based solely on the sequence and environmental information? Inverse protein-folding problem: Is it possible to design a polypeptide sequence which will adopt a given structure under certain environmental conditions?
For example, if two systems of ideal gases are in joint thermodynamic equilibrium across an immovable diathermal wall, then P 1 V 1 / N 1 = P 2 V 2 / N 2 where P i is the pressure in the ith system, V i is the volume, and N i is the amount (in moles, or simply the number of atoms) of gas.
A problem is complete for a complexity class if every problem in the class reduces to that problem, and it is also in the class itself. In this sense the problem represents the class, since any solution to it can, in combination with the reductions, be used to solve every problem in the class. However, in order to be useful, reductions must be ...
Since each element of X belongs to a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by ~, each element of X belongs to a unique equivalence class of X by ~. Thus there is a natural bijection between the set of all equivalence relations on X and the set of all partitions of X.
For example, a construction by transfinite recursion frequently will not specify a unique value for A α+1, given the sequence up to α, but will specify only a condition that A α+1 must satisfy, and argue that there is at least one set satisfying this condition. If it is not possible to define a unique example of such a set at each stage ...
If M is a transitive model, then ω M is the standard ω. This implies that the natural numbers, integers, and rational numbers of the model are also the same as their standard counterparts.