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

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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

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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) ? ? ?, ?
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• 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

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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(?))
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• 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

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• 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)

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

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• 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

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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)

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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
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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???

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• 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

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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
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• 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
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• 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
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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?
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• ?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

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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
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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,

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Exercise 2-4
1. ???? an ?????,??an 2(-3)n5n? (a) a0
(b) a1 (c) a4
6. ??????????? (a)???10??,??????????3?
(e)???????1?2,?????????????
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• 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,

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• 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 ??
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• 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

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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,
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• ? 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)
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• Example 14.
• Table 2. Some useful summation formulae

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

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

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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)