Step Functions

Step functions can be seen as generalizations of characteristic functions.

Step functions can be expressed as b = {b sub a such that a has multiple definitions)

For example consider a set M equal to a coordinate [1,5] expressed on multiple sets Bsub1=[1,2]
Bsub2=[2,4]
Bsub3=[4,5]

Now let all the sub numbers be defined on the y axis as follows
Ysub(1)=3
Ysub(2)=4
Ysub(3)=2

The resulting graph would be a series from 1 to 5 on the horizontal x axis (set A) with a line going from 1 to 2 at point 3 on the y axis, and another line going from 2 to 4 at point 4 on the y axis, and then another line completing the "step" to 5 from 4 to 5 at point 2 on the y axis.

Characteristic Functions

Characteristic functions define the functions of sets and the functions of all object outside the set.

Take for consideration the set A that is part of the Universe. Then the characteristic function can be seen as

f(x)={1 if x is an element of A, and 0 if x is an element of everything but A, (or U-A)}.