Functional relationship

The relation A Ì X ´ Y is called Functional if all its elements (ordered pairs) have different first coordinates. In other words, each element X Î X such that ( x, y ) Î A corresponds to one and only one element Y Î Y .

Obviously, for the functional relation A, each section along X from X contains at most one element. If X is not in the domain D O ( A ) of this relation, then the section along X is empty. If the section along any element of X contains one and only one element, then the functional relation is Everywhere defined .

The matrix of the functional relation contains in each column no more than one single element. The elements X Î X that are not in the domain of definition correspond to the zero column in the matrix.

For example, let X = { X1, x2, x3, x4, x5, x6 } and Y = { Y1, y2, y3 }.

Functional relationship A = {( X1, y1 ), ( X2, y2 ), ( X3, y1 ), {( X5, y3 ), ( X6, y1 )}. In the matrix, the fourth column is zero. Any functional relation can be considered as a function. In this case, the first coordinate X of an ordered pair ( X, y ) Î A is the argument (variable), and the second Y is the image (value) of the function. Normal recording Y = F ( X ) corresponds to the ratio X F Y or ( x, y ) Î F .

So, for any functional relation A, we can define the function F associated with this relation . But the A-1 relation symmetrical to it may not be a function. For our example, the symmetric relation A-1 is: A-1 = {( U1, x1 ), ( U1, x3 ), ( U1, x6 ), ( U2, x2 ), ( Y3, x5 )} and the function does not is, since its elements do not have different first coordinates.

If instead of numerical ones we consider sets of any kind of nature, then we come to the most general notion of a function.

Let M and N be two arbitrary sets. It is said that on the M function is defined the F , taking the value of the of N , if each element of X the Î M mapped to one and only one element have the Î of N . To record this fact using the following symbols F : M ® N .

Thus, the function is a mapping of X in a plurality of Y .

The main properties of mappings are set:

- the pre-image of the sum of two sets is equal to the sum of their pre-images

F-1 ( A B ) = f-1 ( A ) F-1 ( B );

- the pre-image of the intersection of two sets is equal to the intersection of their pre-images

F-1 ( AB ) = F-1 ( A ) F-1 ( B );

- the image of the sum of two sets is equal to the sum of their images

F ( A B ) = F ( A ) F ( B ).

These properties remain valid for the sums and intersections of any (finite or infinite) number of sets.

Note : The image of the intersection of two sets, generally speaking, does not coincide with the intersection of their images.

For example , suppose the mapping in question is the projection of a plane onto the OX axis . Then the segments

0 £ X £ 1, Y = 0

0 £ X £ 1, y = 1

Do not intersect, and their images are the same.