Mathematical Techniques for Computer Science (COMP11120)
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Lecture Notes on Comparing Sets (COMP11120)
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Course
Mathematical Techniques for Computer Science (COMP11120)
Institution
The University Of Manchester (UOM)
Enhance your understanding of set theory with these detailed lecture notes for COMP11120, focusing on comparing sets. Covering key topics such as set equality, subset relations, and power sets, these notes provide clear explanations and practical examples to help you master the concepts of comparin...
Mathematical Techniques for Computer Science (COMP11120)
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Comparing Sets
Definition : Size of set S is less than or equal to size of set T Definition : A set S is infinite iff there is SES
iff there is an injection from S to
T
.
and f : > S'
5- is injective .
Example : IN and Example :
I is infinite
I # Let S =
1 and S =
2 (o]
: !
3 > 3 I I > I <03
2 2 : M
>
S
N + 1 X20
1 > 1 3n XI "
x else
O > O 2
-
1 1 -
O
fiN > 2
-
-
2 +- 7
-
3 -
1 ?
25
-
:
-
3?
Definition : Two sets S andT have the same size Proposition : A set S is infinite iff there is
f : S <S that is injective and
iff size of S size
of T not surjective.
and size of T & Size
of S
Example : IN and Proposition If : a set S is infinite ,
it is at least as big as IN.
IN is minimal infinite set !
I # I
: ! : know have fi5 >
-
S ,
f is injective , not surjective.
M >
4 : 4 Have seS such that Use S .
fs' = S
3 > 3 93
2 2 : 2
Prove: Want > S
IN- injective
>
is
9 g
:
,
Note : Much easier to establish sets have the same size than show there is a W
bijection num >
from = S
- n -
m = 0
>
- n = m
Definition : A set is countable iff its size is at most that of IN
A set is uncountable iff it is not countable
Proposition :
If the size of S is at least the size of IN , then S is infinite.
A set is countably infinite iff it is countable and infinite . Our notion of size , infinity makes sense
Why are these notions important for computer science ?
Set of programs in
your favorite language > Countable
Examples
Set of functions from IN to IN
>
Uncountable
Countable Uncountable
·
N . . Q ·
R ·
SF : IN <
Ny
·
Powerset of IN
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