For example, let N1 be the {R} structure whose domain is the set of natural numbers and which interprets 'R' as the relation smaller than.
M is a trivial matrix whose domain is the set of formulas of the system, whose designated elements are the theorems of the system, and whose operations are the connectives themselves.
To overcome the problem of suitable direction selection, the use of the Peano Hilbert space filling curves (SFCs) is proposed, which produces a continuous and unique function whose domain is the unit interval [0,1].
Using Zernike moments bears some immediate advantages: they provide a direct way of encoding images whose domain is the unit disc and they can provide rotational invariance [17], which is ideal considering how the sensor works.
It is natural to understand the unraveled form is as an infinite sequence; standardly, infinite sequences are taken to be functions whose domain is the set N of natural numbers.
We might, then, think of the intension of "\(1 + 4\)" as a partial function on states, whose domain is the set of states in which the instructions inherent in "\(1 + 4\)" have been executed, and mapping those states to 5.
For example we can require that the strategy function telling ∃ what to do at a particular step is a function whose domain is the family of possible choices of ∀ at just his first and second moves; this is a way of expressing that ∃ doesn't know how ∀ chose at his third and later moves.
Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions.
The nation-state, which grew up alongside the First and Second Industrial Revolutions, and provided the regulatory mechanism for managing an energy regime whose reach was the geosphere, is ill suited for a Third Industrial Revolution whose domain is the biosphere.
Then, as noted in section 1, the Transitive Submodel Theorem says that if we start with any transitive model of ZFC, then we can find a transitive model whose domain is countable (indeed, we may assume that this countable model is a submodel of the model with which we started).
In 1915, Leopold Löwenheim proved that if a first-order sentence has a model, then it has a model whose domain is countable.[3] In 1922, Thoralf Skolem generalized this result to whole sets of sentences.
