Mathematical Foundation of Computer Science

DHARM

28 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE

Example 2.3. Add the numeric functions

an =

0for0n5

32forn6n

≤≤

+≥

R

S

T − and bn =

13 for0n2

n5 forn3

−≤≤n

+≥

R
S
T
Sol. Let an + bn = cn, then cn will be given as,

cn =

013 13 0 2

055 35

32 53 7 6

+− =− ≤ ≤

++=+ ≤≤

++ += + + ≥

R

S

|

T|

−−

nn

nn

n

nn n

nnn

for

for

for

Example 2.4. Add the numeric functions

an =^0 for 0 n 2

25forn3n

≤≤

+≥

R

S

T

− and bn^ = 32 for0n1n2 forn2

−≤≤n

+≥

R
S
T
Sol. Let an + bn = cn, then cn will be given as,

cn =

032 32 0 1

0224 2

25 22 7 3

+− =− ≤≤

++= =

++ += + + ≥

R

S

|

T|

−−

nn

nn

n

n

nnn

for

for

for

()

Addition property of numeric function can also be applied between two/ more numeric
functions. Lets we have numeric functions a 1 , a 2 , .......an then their addition a 1 + a 2 +...... + an
will also a numeric function whose value at n is equal to the sum of values of all the numeric
functions at n. For example,

an =

10

21

02

for

for

for

n

n

n

=

=

≥

R

S

|

T|

bn =

002

23

for

for

≤≤

≥

R

S

T

n

n n and cn =

10

01

for

for

n

n

=

≥

RS

T

then an + bn + cn =

1012 0

2002 1

0000 2

02 02 3

++= =

++= =

++= =

++= ≥

R

S

||

T

|

|

for

for

for

for

n

n

n

nnn

2.2.2 Similar to additions of numeric functions, multiplication of numeric functions also
returns a numeric function. Let an and bn are two numeric functions then its multiplica-
tion (an ∗ bn) will be a numeric function and its value at n will be the multiplication of
values of numeric functions at n.(Multiplication property of numeric functions).
For example, the numeric functions

an =

002

23

for

for

≤≤

≥

R

S

T

n

n n and bn = 2

n for n ≥ 0

then an * bn =

02 0 0 2

22 2^13

*

*

n

nn n

n

n

=≤≤

=≥

R

S

T

+

for

for

Example 2.5. The multiplication of numeric functions given in example 3.4, will be

an * bn =

0 * (3 2 ) 0 for 0 n 1

0*(n 2) 0 for n 2

(2 5) * (n 2) for n 3

n

n

−= ≤≤

+= =

++ ≥

R

S

|

T|

−