**Formal Definition**

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":

**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 ...

### Famous quotes containing the words definition and/or formal:

“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by *definition* things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”

—Elaine Heffner (20th century)

“On every *formal* visit a child ought to be of the party, by way of provision for discourse.”

—Jane Austen (1775–1817)