Formal DefinitionSee also: Jordan measure
An approach to defining what is meant by "area" is through axioms. "Area" can be defined as a function from a collection M of special kind of plane figures (termed measurable sets) to the set of real numbers which satisfies the following properties:
- For all S in M, a(S) ≥ 0.
- If S and T are in M then so are S ∪ T and S ∩ T, and also a(S∪T) = a(S) + a(T) − a(S∩T).
- If S and T are in M with S ⊆ T then T − S is in M and a(T−S) = a(T) − a(S).
- If a set S is in M and S is congruent to T then T is also in M and a(S) = a(T).
- Every rectangle R is in M. If the rectangle has length h and breadth k then a(R) = hk.
- Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. S ⊆ Q ⊆ T. If there is a unique number c such that a(S) ≤ c ≤ a(T) for all such step regions S and T, then a(Q) = c.
It can be proved that such an area function actually exists.
Read more about this topic: Area
Other articles related to "formal definition":
... The above controller for crosswalk lights can be modeled by an atomic SP-DEVS model ... Formally, an atomic SP-DEVS is a 7-tuple where is a finite set of input events is a finite set of output events is a finite set of states is the initial state is the time advanced function which defines the lifespan of a state where is the set of non-negative rational numbers plus infinity ...
... In typical usage, the formal definition of O notation is not used directly rather, the O notation for a function f(x) is derived by the following simplification rules If f(x) is a sum of several terms, the one with ... One may confirm this calculation using the formal definition let f(x) = 6x4 − 2x3 + 5 and g(x) = x4 ... Applying the formal definition from above, the statement that f(x) = O(x4) is equivalent to its expansion, for some suitable choice of x0 and M and for all x > x0 ...
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