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Sets Lecture notes
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Lecture notes of 43 pages for the course Logic and Sets at VU (Sets Lecture notes)
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Sets lecturesweek 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
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