Title: Chapter 2 Basic Structures : Sets, Functions, Sequences, and Sums
1Discrete Mathematics
- Chapter 2 Basic Structures Sets, Functions,
Sequences, and Sums
???? ????? ???
22-1 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
4Exercise 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) ? ? ?
5Exercise 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
7Exercise 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)
10Exercise 2-1
23. ?A a, b, c, d?B y, z,C1,2?? (a)
A?B (b) B?A (c) B?C?A
112-2 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
13Exercise 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)
142-3 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
15g(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
17Exercise 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
21Exercise 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
23Exercise 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
242.4 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,
25Exercise 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
29Exercise 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
-
32Exercise 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. -
34Example 18. Show that the set of odd positive
integers is a countable set.
35Example 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)