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Title: Chapter 2 Basic Structures : Sets, Functions, Sequences, and Sums


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.
  • x ? A ??A?????,?x?A??????A x, y, z
  • ????????,??????,??????????,????????Q x
    x??? ? Q x x ???10???? ?Q x x
    ???,3ltxlt100
  • ? ?????

3
  • Example 5 ???????
  • N 0,1,2,3,, the set of natural number
    (???)(???????0?????)
  • 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?????)
  • Def 3 A,B sets. AB iff ?x (x ?
    A ? x ? B)
  • Def 4 A ? B iff ?x (x ? A ? x ? B)
    (A?B????)?? A ? B ??A ? B ? A ? B

4
Exercise 2-1
5. ??2??????????? (a) x ? R x ???1???
(b) x ? R x ?????? (c) 2, 2 (d)
2, 2 (e) 2, 2, 2 (f)
2
7. ??????????? (a) 0 ? ?
(b) ? ? 0 (c) 0 ? ? (d)
? ? 0 (e) 0 ? 0 (f)
0 ? 0 (g) ? ? ?
5
Exercise 2-1
8. ??????????? (a) ? ? ?
(b) ? ? ?, ? (c) ? ? ?
(d) ? ? ? (e) ? ? ?, ?
(f) ? ? ?, ?
6
  • 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 A1,2P(A) ?, 1, 2, 1,2
  • Example 13 S 0,1,2P(S) ?, 0, 1,
    2, 0,1, 0,2, 1,2, 0,1,2

7
Exercise 2-1
17. ????????(cardinality)??? (a) a (b)
a (c) a, a (d) a, a, a, a
21(??). ??????,??a?b?????? (a) P(a, b) (b)
P(?) (c) P(P(?))
8
  • Def 9 A, B sets. The Cartesian Product
    (????) of A and B, denoted by A?B, is the set
    A?B (a, b) a ? A and b ? B
  • Example 16 A 1,2 , B a, b, c
  • A?B (1, a), (1, b), (1, c), (2,a), (2,b),
    (2,c)
  • Note A?B A?B

9
  • Def 9 A1, A2, , An sets. The Cartesian
    Product (????) of A1, A2, , An, denoted by
    A1?A2??An, is the setA1?A2? ?An (a1, a2,
    , an) ai ? Ai,
    where i1, 2, , n
  • Example 18 A 0,1 , B x,y, Ca,b,c
  • A?B?C (0,x,a), (0,x,b), (0,x,c),
    (0,y,a), (0,y,b), (0,y,c),
  • (1,x,a), (1,x,b), (1,x,c),
    (1,y,a), (1,y,b), (1,y,c)

10
Exercise 2-1
23. ?A a, b, c, d?B y, z,C1,2?? (a)
A?B (b) B?A (c) B?C?A
11
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(???)

12
  • Def 6,7 A1 , A2 , , An sets
  • Let I 1,3,5 ,
  • Def (??31) 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

13
Exercise 2-2
14. ?A-B 1,5,7,8, B-A 2,10,??AnB
3,6,9? ????A?B?
32. ?? 1,3, 5?1, 2, 3?????(symmetric
difference)?
  • 26. ?ABC???,?????????????(a) A n (B?C) (b) A n
    B n C(c) (A-B)?(A-C)?(B-C)

14
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.
  • Example

f(a)1f(b)1f(b)2f(g)3
f(a)1f(b)?f(g)2
Not a function
Not a function
15
g(a)1g(b)2g(g)1
f(a)1f(b)4f(g)2
a function
a function
  • Def 2 (? f A?B ??,???)
  • A domain (???) of f , B codomain (???) of
    f
  • f (a) 1 , f (b) 4 , f (g) 2
  • 1??a?image (??, ????), a??1?pre-image(??,
    ?????)
  • 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???

16
  • Def 3 ? f1 ? f2 ? A ? R ???,?f1 f2 ? f1 f2??
    A ? R ???,?????
  • ( f1 f2 )(x) f1(x) f2(x)
    (f1 f2)(x) f1(x) f2(x)
  • Example 6 Let f1 R ? R and f2 R ? R such
    that
  • 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

17
Exercise 2-3
18
? ??????????
  • Def 5 A function f is said to be one-to-one
    (???), or injective (??), iff f (x) ? f (y)
    whenever x ? y.
  • Example 8

? 1-1
?? 1-1 , ? g(a) g(d) 4
19
  • 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.
onto
not onto
20
  • Def 8 The function f is a one-to-one
    correspondence (???????), or a bijection (??), if
    it is both 1-1 and onto.
  • ?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.

1-1 , not onto
not 1-1, onto
1-1 and onto
21
Exercise 2-3
12, 13, 19. ?????? ?Z???Z??????????????????
(a) f (n) n-1 (b) f (n) n21 (c) f
(n) n3 (d) f (n) ?n/2?
22
  • ?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

23
Exercise 2-3
27. ?f (x) ?x2/3???????,?? f (S)? (a) S
-2, -1, 0, 1, 2, 3 (b) S 0, 1, 2, 3, 4,
5 (c) S 1, 5, 7, 11 (d) S 2, 6,
10, 14
24
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 1, 2, 3,
  • ? a1 1, a2 1/2 , a3
    1/3,
  • Example 2. bn , where bn (-1)n, n 0, 1,
    2,
  • ? b0 1, b1 -1 , b2
    1,

25
Exercise 2-4
1. ???? an ?????,??an 2(-3)n5n? (a) a0
(b) a1 (c) a4
6. ??????????? (a)???10??,??????????3?
(e)???????1?2,?????????????
26
  • Example 5. ????????,??????(a) 1, 1/2, 1/4,
    1/8, 1/16(b) 1, 3, 5, 7, 9(c) 1, -1, 1, - 1, 1
  • Sol
  • (a) a0 1, a1 1/2, a2 1/22, a3 1/23,
    a4 1/24,
  • ? an 1/2n, n 0, 1, 2, 3,
  • (b) a0 1, a1 3, a2 5, a3 7, a4 9
  • ? an 2n1, n 0, 1, 2, 3,
  • (c) a0 1, a1 -1, a2 1, a3 -1, a4 1
  • ? an (-1)n, n 0, 1, 2, 3,

27
  • 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, n 1, 2,
    3,

????????? a0 ?? a1 ??? an ???,
???????????? n ? 0 ?? 1 ??
28
  • 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

29
Exercise 2-4
9. ??????????????????,????????,???????? (c)
1, 0, 2, 0, 4, 0, 8, 0, 16, 0, (d) 3, 6,
12, 24, 48, 96, 192, (e) 15, 8, 1, -6, -13,
-20, - 27, (f) 3, 5, 8, 12, 17, 23, 30,
38, 47, (g) 2, 16, 54, 128, 250, 432, 686,
(h) 2, 3, 7, 25, 121, 721, 5041, 40321,
30
  • ? Summations (??,????)
  • Here, the variable j is call the index of
    summation, m is the lower limit (??), and n is
    the upper limit (??).

Example 10.
122232425255 Example 13. (Double
summation)
31
  • Example 14.
  • Table 2. Some useful summation formulae

32
Exercise 2-4
13. ????????? (a)
(b) (c) (d)
33
  • ?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.

34
Example 18. Show that the set of odd positive
integers is a countable set.

35
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
(??,? ?? ,? ??)
?Note. R is uncountable. (Example 21)
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