Lecture Notes on Induction for the Natural Numbers and Introduction to Relations and Partial Functions (COMP11120)

Induction for The Natural Numbers, Introduction to Relations and Partial Functions

Recursive Definition for natural numbers : I

Can now
formally define operator Property i =
n(n n E I

aie IN
O
0 (0 + 1)
base
.




case
o = 0 7
2

base case a = [ * ai = a1
i=1 1
n+

n
-




Step case i = i +
i= 0


a
L

Step case = + an e

1)
nin n
= +
+




=
n2 + n + 2n + 2

2


S
= (n+ 1)(n + 2)
2




Functions given by recursive specifications
These arise when calculating the complexity of recursive algorithms.
, eg .,

Example : Find a functionf that satisfies base case fo = 2 Example : Find a functionf that satisfies

Step case fin + 1) =
2fn-1 base case fo = 1


base case fi = 4




-123
Step case f(n + 2) =
4fn +
3f(n + 1)
Guess >
-


fn = 2 + 1


1 24 81632 n O 1 2 3 4 5
Gress fr = yh
16 64256512
In 1 ↑



base case fo = 2

base 1
=
20 + 1 case fo =




40
2 fn
=

-1
Step case finall =




ind hyp! 4
=
2(2n + 1) -

1 base case fr =




=
24
+1
+ 2 -

1
= m

=
2n
+1
+ 1 Step case final =
4 fn + 3 f(n 1) +



= 4(yn) + 3)yn
+
1) -
ind . hyp!
+1
1)
+
= yn + 3(yn

yn 1(1 3)
+
= +


*2
=
yn




Relations
Relations are a
way of connecting elements of sets
. Example : Consider the relation R from [W ,
X
,
Y , 2) to [10 , 15 , 203 that relates

Relations between two or more sets are used in databases
w
· to 10 , 13 and 20


Crolational databases) for example to capture student records
· I to 10 and
,
with entries such as

2 to 10 and 20
(name , id number degree type degree name).
·

, ,




W 10
Notation

A relation R from a set S to set T is a subset of
Z 15

SxT , i . e . RESxT Y 20


This is sometime written as 2

R : S 1 > T

We say
in fix notation ! R is a subset
of pairs from CW ,
K
, y, 23 x 410 ,
15 , 203
(s t) E R -
Rt &(w y
or
,
R = ,
10) ,
(2 , 15) , (W , 20) , (X, 10) , (2 , 10) , (2 , 20)

or S ~t
↑
symbol to
denote relation