Mathematical Techniques for Computer Science (COMP11120)
Class notes
Lecture Notes on Induction for the Natural Numbers and Introduction to Relations and Partial Functions (COMP11120)
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Course
Mathematical Techniques for Computer Science (COMP11120)
Institution
The University Of Manchester (UOM)
Strengthen your understanding of mathematical induction and foundational concepts in relations and partial functions with these detailed lecture notes for COMP11120. Covering topics such as the principles of induction, properties of natural numbers, and an introduction to relations and partial func...
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
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