离散数学第一章

The
operations of intersection and union can be extended to three or more sets.
={ x | xA or xB or xC } An ∩B∩C ={ x | xA and xB and xC }
A∪B∪C
7
1.1 Sets and Subsets
How
By
to describe a set?
specifying a property that the elements of the set have in common
For
any proposition P(x) over any universe of discourse, { x | P(x) } is the set of all x such that P(x). { x | x is an positive integer} { x | x is a real number and x2=7 }
1.1 1.2 1.3 1.5
Sets and Subsets
Operations on Sets
Sequences
Matrices (self-study)
17
1.2 Operations on Sets
Let
A and B be two sets. We can make new sets from old sets using the following operations:

18
1.2 Operations on Sets
= { x | xA and xB } A∪B = { x | xA or xB } A-B = { x | xA and xB } A = U -A AB = (A-B)∪(B-A)
A∩B
19
1.2 Operations on Sets
For
finite A, | P (A) | = ?
14
1.1 Sets and Subsets
Two
sets are equal if they contain exactly the same element A=B if and only if AB and BA Proper Subsets
10
1.1 Sets and Subsets
Subset
Given
and Superset
two sets A and B . if every element in A is an element of B, we say that A is a subset of B or that A is contained in B, and B is a superset of A, and we write AB
8
1.1 Sets and Subsets
How
By
to describe a set?
specifying a property that the elements of the set have in common
Z+
= { x | x is a positive integer } Z = { x | x is an integer } Q = { x | x is a rational number } R = { x | x is a real number }
U
21
1.2 Operations on Sets
Venn
diagrams
U
A
B
22
1.2 Operations on Sets
Venn
diagrams
B
AB
A
Fra Baidu bibliotek
24
1.2 Operations on Sets
Venn
diagrams
B
AB
A
25
1.2 Operations on Sets
AB and A≠B Denoted by AB
If
15
1.1 Sets and Subsets
The
universal set U
Containing
all objects for which the discussion is useful.
16
Chapter 1 Fundamentals
A -B
A ∩B
29
1.2 Operations on Sets
Venn
diagrams
A
B
AB
A ∩B
30
1.2 Operations on Sets
Venn
diagrams
A
U B
A
A ∩B
31
1.2 Operations on Sets
= {0, 1, …, 9}, A = {0, 1, 2, 3}, B = {1, 3, 5, 7, 9} A∪B = {0, 1, 2, 3, 5, 7, 9} A∩B = {1, 3} A-B = {0, 2} A = {4, 5, 6, 7, 8, 9} B = {0, 2, 4, 6, 8} AB = {0,2,5,7,9}
We
4
1.1 Sets and Subsets
How
By
to describe a set?
listing all its elements By specifying a property that the elements of the set have in common
5
1.1 Sets and Subsets
Venn
diagrams
A
U
A
26
1.2 Operations on Sets
Venn
diagrams
A
B
A∩B
27
1.2 Operations on Sets
Venn
diagrams
A
B
A∪B
A ∩B
28
1.2 Operations on Sets
Venn
diagrams
A
B
U
32
1.2 Operations on Sets
U
B
5 3
4 0 A 6
7
9
1
2
8
33
1.2 Operations on Sets
Algebraic
Properties of Set Operations
6
1.1 Sets and Subsets
How
By
to describe a set?
listing all its elements
Sets
are inherently unordered: {a, b, c} = {a, c, b} = {b, a, c} ={b, c, a} = {c, a, b} = {c, b, a} All elements are distinct (unequal); multiple listings make no difference! {a, a, b, a, b, c, c, c, c} = {a, b, c}
13
1.1 Sets and Subsets
Power
Let
Set
A be a set. Then set of all subsets in A is called the power set of A and is denoted as P (A).
A={a,
Example:
b, c} P (A)={ ø , {a, b}, {a, c}, {b, c}, {a}, {b}, {c}, {a, b, c} }
i 1 n
Ai Ai
i 1
20

1.2 Operations on Sets
= {0, 1, …, 9}, A = {0, 1, 2, 3}, B = {1, 3, 5, 7, 9} A∪B = {0, 1, 2, 3, 5, 7, 9} A∩B = {1, 3} A-B = {0, 2} A = {4, 5, 6, 7, 8, 9} B = {0, 2, 4, 6, 8} AB = {0,2,5,7,9}
Intersection of two sets: A∩B Union of two sets: A∪B Difference of two sets: A-B or A\B ( Complement of B with respect to A ) Complement of one set: A Symmetric difference of two sets: AB = (A\B)U(B\A)
Subset
let
and Superset
versus
Example:
Since
A be a set and let B={A, {A}}.
A and {A} are elements of B, we have AB and {A}B. It follows that {A}B and {{A}}B. However, it is not true that AB.
How
By
A
to describe a set?
listing all its elements
set is canonically described by listing its elements between “{” and “}” {a, b, c} is the set of whatever 3 objects are denoted by a, b, c.
3
1.1 Sets and Subsets
write xA to say that element x belongs to the set A Elements are named with small letters (a, b, c, x, y, z, etc.) while capital letters are reserved to sets (A, B, C, E, F, S, U etc.) The set with NO elements is called the empty set and denoted by (read as phi)
Sequences
Matrices (self-study)
2
1.1 Sets and Subsets
What

is Set?
A set is an well-defined collection of objects. e.g. the set of all real numbers, the set of all chairs in this room, the set of all girls in this room, etc. Individual items in a set are called elements or members. NOTE: there is no relationship between different elements of a general set
Chapter 1 Fundamentals
BAO Peng baopeng@bjtu.edu.cn Monday, September 14, 2015
Chapter 1 Fundamentals
1.1 1.2 1.3 1.5
Sets and Subsets
Operations on Sets
11
1.1 Sets and Subsets
Subset
and Superset
Example:
A={1, 3} is contained in B={9, 7, 5, 3, 1}
For
any set A, we have øA and A A .
12
1.1 Sets and Subsets
9
1.1 Sets and Subsets
We
designate by card(A) or |A| the cardinality of a set A, which is the number of distinct element in A. Example: card({a,b,2,a,☻}) = ? A set A is finite if card(A) <∞ otherwise it is infinite.