Mathematical Foundation of Computer Science

DHARM

DISCRETE NUMERIC FUNCTIONS AND GENERATING FUNCTIONS 27

Since 1322.5 + 1322.5 (1 + 18/100)5 = 1322.5 + 1322.5 (1.18)5 = Rs 4348.06
Therefore, An = 4348.06 + 4348.06 (1 + 10/100)n for 8 ≤ n
Below in this section we will see couple of examples that concisely represent the nu-
meric functions. In general we use convention small letters to represent the numeric func-
tions.

Example 2.1.

• an = 3 n^2 + 1 for n ≥ 0
  (A simple expression of numeric function for every n is defined)


• bn =^503
  04

nn

n

for

for

≤≤

≥

R

S

T

(Here two different numeric expressions are defined for different domain sets)

• cn =

nn

nnn

nnn

−≤≤

+≥ =

>≠

R

S

|

T|

307

3880

3780

for

for and

for and

%
/%
(Here different numeric expressions are defined for different domain sets)
Example 2.2. Consider an example of airplane, lets hn be the altitude of the plane (in thousand
of feet) at nth minute. The plane land off after 10 minutes on the ground and ascend to an
altitude 10000 feet in 10 minutes at a uniform speed. Plane starts to descend uniformly after
one hour of flying and after 10 minutes it lands. Thus we have different numeric functions that
measure the altitude for different slices of the plane journey.
Sol.
0 for 0 ≤ n ≤ 10 [upto 10 minutes plane remain on ground]
n – 10 for 11 ≤ n ≤ 20 [Plane ascend uniformly at 1000 feet per
minute so altitude is proportional to time
or (n – 10 )]
10 for 21 ≤ n ≤ 80 [next 1 hour plane will remain on altitude 10]
hn = 90 – n for 81 ≤ n ≤ 90 [next 10 minutes plane returns to ground with
uniform speed, after first minute of time plane
will be at height 9000 feet and so on...]
0 for n > 90 [Plane remain in ground so no altitude]

2.2 Properties of Numeric Functions.....................................................................................

In this section we shall discuss the behavior of numeric functions over unary and binary

operations.

2.2.1 Let an and bn are two numeric functions then its addition (an + bn) given by cn is also a

numeric function. The value of cn at n will be the addition of values of numeric functions

an and bn at n. (Addition property of numeric functions)

For example, an =

002

23

for

for

≤≤

≥

R

S

T

n

n n and bn = 2

n for n ≥ 0

then cn = an + bn = 02 2^02

2221 3

+= ≤≤

+= ≥

R

S

T

+

nn

nnn

n

n

for

for

R

S

|

|

|

|

||

T

| | | | | |