Sets Lecture notes

Sets lectures

week 1 Sets
sets definition and notation:




DEFINITION AND NOLATION OF SETS
-




a set is an unordered collection of elements .




examples
days of the we e k : =
[Mon ,
Tue ,
Wed ,
Thu ,
Fri ,
Sat , Sund
A : =
51 ,
2 ,
33
digits
: =
50 ,
1 , .... 93


N : =
20 ,
1 ,
2
,
3
, . . .
4 natural numbers


I G Y
: =
2 -1 0 1 2
integer numbers
-




.... , , , , ,
. . .




prototype notation


multiples of 2 :=
G2K : K is a natural numbery =
50 ,
2
, ,
4 . . .
/
month : =
EX :
X is a months


ELEMENT AND SUBSET
element


at A a is an element of the Set A .




a A a is not an element of the set A
.




subset

A = B all elements of A are elements of B
.

A B at least one element of A is not in .
B




examples

4 [1 ,
2 ,
3 . 43
5 [1 ,
2 .
3 , 43


92 33 .
=
41 ,
2 ,
3 .
44
[2 53 * 41 . ,
2 .
3 , 43

SET EQUALITY EMPTY SET
definition of set equality
A =
B = A & B and BEA




examples

92 ,
2 ,
3 , 43 =
94 ,
3 ,
1 , 27 =

24 ,
3 ,
3 ,
2
,
1
, 24

, 51 43 + 92
,
2
,
3 , ,
3
,
4 , 53


definition of empty set



① or 53 is the cunique) set t h at has no elements .




number of (distinct) e l e m e n ts



# [4 ,
3 ,
3
,
2 ,
1
, 23 =
#91 ,
2 ,
3 . 47
=
4



# 0 =
0




EXERCISES
1. all subsets o ...
f



573 - 0 , 513

51 23 .
20 ,
313 , 523 ,
31 23 ,




.
2 are the following true or false ?


21 ,
3 ,
5 , 73 =97 ,
6 ,
5 ,
4 , 3 ,
2
,
6, 04
↳ FALSE !




91 ,
3 ,
5 , 73 -
27 ,
4
,
1
,
6 ,
5
,
4 ,
4
,
33
↳ TRUE



① = O ,
in fact I any set .




fundamental set operations
UNIVERSE U
all our sets are subsets of a universe .
U

U


U : all four-letter words
A

A: words with "a"




COMPLEMENT
complement of A

Y U : all four-letter words

A: words with "a"
A




A : words with no "a"

A : =
[x = U :
x4 A3

,UNION U

union of A and B


W U : all four-letter words

A: words with "a"
A B

3 words with "b"




AU B : words With "a" or "b"

inclusive or




AUB : =
Gxt4 :
XA or X =
By


INTERSECTION
intersection of A and B


U : all four-letter words

= D A: words with "a"

3 words with "b"




A 1 B : words With "a" and "b"

AlB : =
GxH : XEA and x =
B]


MINUS
A minus
U all four-letter words



·
:




A: words with "a"

3 words with "b"




A B :
words with "a" but no "b"

AlB : =
GXA :
#By = A1 B'



SYMMETRIC DIFFERENCE S

symmetric difference of A and B


U : all four-letter words



D A:


3
words


words with
with

"b"
"a"




A 1 B : words with "a" or "b" ,
but not both .




exclusive or


A1 B
: =
(AUB)n(A1B)
=
(A(B)U(B(A)

, EXERCISES
given :




A : =
50 ,
1
,
2 ,
33

B : =

G2 ,
3
,
4
,
53

UE N




A U B =
50 ,
1 ,
2
,
3 ,
4
,
53
A 1B =
92 3) ,




A =
Ex = N :
x -
3) =
24 ,
5 ,
6
, 7 .... %

B =
(x = N : x # 22 ,
3 ,
4 , 533 =
20 ,
1 ,
6
, 7 ,
0 ,
9 ,
10 .... 3


AlB =
20 13 ,




B1A =

54 53 ,




A
-
B =
50 ,
7
,
4 , 53


venn diagrams ona partitions
VENN DIAGRAM
a venn diagram is an abstract visualization of relations between sets .


W


2 Sets regions
>
-
↑
A B




W

A B
3 Sets >
-
8 regions

E




every region in a venn diagram can be empty or non-empty



DE MORGAN'S A'n B' =
(AUB)

D1
Al

A




J :
A
↑
&




D
complement




B
AU
(AUB)'

B