Discrete Mathematics

1
Discrete Mathematics

• Chapter 2 Basic Structures Sets, Functions,
  Sequences, and Sums

???? ????? ???
2
2-1 Sets

• Def 1 A set is an unordered collection of
  objects.
• Def 2 The objects in a set are called the
  elements , or members of the set.
• Example 5 ???????
• N 0,1,2,3, , the set of natural number
  (???)
• Z ,-2,-1,0,1,2, , the set of integers (??)
• Z 1,2,3, , the set of positive integers
  (???)
• Q p / q p ? Z , q ? Z , q?0 , the set
  of rational numbers (???)
• R the set of real numbers (??)
  (??????1.234?????)

3

• Def 4 A ? B iff ?x , x ? A ? x ? B ?? A ? B
  ??A ? B ? A ? B
• Def 5 S a finite set
• The cardinality of S , denoted by S , is
  the number of elements in S.
• Def 7 S a set
• The power set of S , denoted by P(S) , is the
  set of all subsets of S.
• Example 13 S 0,1,2
• P(S) ?, 0 , 1 , 2 , 0,1 , 0,2 ,
  1,2 , 0,1,2
• Def 8 A , B sets The Cartesian Product of A
  and B , denoted by A x B , is the set A x B
  (a,b) a ? A and b ? B

4

• Note. A x B A.B
• Example 16
• A 1,2 , B a, b, c
• A x B (1,a) , (1,b) , (1,c) , (2,a) , (2,b) ,
  (2,c)
• Exercise 5, 7, 8, 17, 21, 23

5
2-2 Set Operations

• Def 1,2,4 A,B sets
• A?B x x ? A or x ? B (union)
• AnB x x ? A and x ? B (intersection)
• A B x x ? A and x ? B (????A \ B)
• Def 3 Two sets A,B are disjoint if AnB ?
• Def 5 Let U be the universal set.
• The complement of the set A , denoted by A ,
  is the set U A .
• Example 10 Prove that AnB A?B
• pf
• ?? Venn Diagram

6

• Def 6 A1 , A2 , , An sets
• Let I 1,3,5 ,
• Def (p.131??) A,B sets
• The symmetric difference of A and B , denoted by
  A?B , is the set
• x x ? A - B or x ? B - A ( A?B ) - ( A
  nB )
• ?Inclusion Exclusion Principle (????)
• A ? B A B - A n B
• Exercise 14, 45

7
2-3 Functions

• Def 1 A,B sets
• A function f A ? B is an assignment of
  exactly one element of B to each element of A.
  We write f(a) b if b is the unique element of
  B assigned by f to a ? A.
• eg.

A
B
A
B
1
a
a
1
ß
ß
2
2
?
3
?
Not a function
Not a function
8
A
B
A
B

• Def (? f A?B ??,???)
• f (a) 1 , f (ß) 4 , f (?) 2
• 1 ??a?image (???) , a??1?pre-image(?????)
• A domain of f , B codomain of f
• range of f f (a) a ? A f (A) 1,2,4
  (??B)
• Example 4 f Z ? Z , f (x) x2 , ? f ?domain
  , codomain
• ?range ?

1
a
a
1
2
2
ß
ß
3
?
?
4
a function
a function
9

• Example 6 Let f1 R ? R and f2 R ? R s.t.
• f1(x) x2 , f2(x) x - x2 , What are the
  function f1 f2 and f1 f2 ?
• Sol
• ( f1 f2 )(x) f1(x) f2(x)
  x2 ( x x2 ) x
• (f1 f2)(x) f1(x).f2(x) x2( x
  x2 ) x3 x4
• Def 5 A function f is said to be one-to-one , or
  injective , iff f (x) ? f (y) whenever x ? y.
• Example 8

f
g
A
B
A
B
1
1
a
a
2
2
b
b
3
3
c
d
4
4
5
c
d
5
? 1-1
?? 1-1 , ? g(a) g(d) 4
10

• Example 10 Determine whether the function f (x)
  x 1 is one-to-one ?
• Sol x ? y ? x 1 ? y 1
• ? f (x) ? f (y)
• ? f is 1-1
• Def 7 A function f A ? B is called onto , or
  surjective , iff for every element b ? B , ?a ? A
  with f (a) b. (? B ??????? f ???)
• Example 11

Note ?A lt B ?,????onto.
11

• Def 8 The function f is a one-to-one
  correspondence , or a bijection , if it is both
  1-1 and onto.
• Examples in Fig 5
• ??? f A ?B
• (1) If f is 1-1 , then A B
• (2) If f is onto , then A B
• (3) if f is 1-1 and onto , then A B.

12

• ?Some important functions
• Def 12
• floor function x x ?????,? x
• ceiling function x x ?????.
• Example 24
• ½ -½
  7
• ½ -½
  7
• Example 29
• factorial function
• f N ? Z , f (n) n! 1 x 2 x x n
• Exercise 1,12,17,19

13
2.4 Sequences and Summations

• ?Sequence (??)
• Def 1. A sequence is a function f from A ? Z
• (or A ? N) to a set S. We use an to
  denote f(n), and call an a term (?) of
  the sequence.
• Example 1. an , where an 1/n , n ? Z
• ? a1 1, a2 1/2 , a3
  1/3,
• Example 2. bn , where bn (-1)n, n ? N
• ? b0 1, b1 -1 , b2
  1,

14

• Example 7. How can we produce the terms of a
  sequence if the first 10 terms are 5, 11, 17,
  23, 29, 35,41, 47, 53, 59?
• Sol
• a1 5
• a2 11 5 6
• a3 17 11 6 5 6 ? 2
• ? an 5 6 ? (n-1) 6n-1

15

• Example 8. Conjecture a simple formula for an
  if
• the first 10 terms of the sequence an are
• 1, 7, 25, 79, 241, 727, 2185, 6559, 19681,59047?
• Sol
• ???????
• ??????????3
• ? ????? 3n ?
• ??
• 3n 3, 9, 27, 81, 243, 729, 2187,
• an 1, 7, 25, 79, 241, 727, 2185,
• ? an 3n - 2 , n ? 1

16

• ? Summations
• Here, the variable j is call the index of
  summation, m is the lower limit, and n is the
  upper limit.

Example 10. Example 13. (Double summation)

17

• Example 14.
• Table 2. Some useful summation formulae

18

• ?Cardinality
• Def 4. The sets A and B have the same cardinality
  (size) if and only if there is a one-to-one
  correspondence (1-1,onto ?function) from A to B.
  
• Def 5. A set that is either finite or has the
  same
• cardinality as Z (or N) is called countable
  (??).
• A set that is not countable is called
  uncountable.

19
Example 18. Show that the set of odd positive
integers is a countable set.

20
Example 19. Show that the set of positive
rational number (Q) is countable.
Pf Q a / b a, b? Z
(Figure 2)
? Z 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 ,
9 Q
Exercise 9,13,17,42
(??,? ?? ,? ??)
?Note. R is uncountable. (Example 21)