?? (1.2? pp. 6~8)

1
????? I-????????-

• ??3??
• ?? ??

2
??(1.2? pp. 68)

• ??(set)
• ????????????(?????)?????
• ?(member)
• ???????

??
?
3
??????????

• A?B?????A?B?????
• A???????B??
• ????? A?B
• A?B??A?B
• A???????B?????
• A?????B?????
• A?B??A?B AB

B
A
B
A
4
??????

• ??????
• 0, 1 dog, cat
• ????????????
• xP(x)
• ?P(x)????????x????
• ???????????
• x?AP(x)
• ?P(x)????????A??x????
• xP(x)??x?A

5
?????(???)

1. ? (union) A?B
2. ???? (intersection) AnB
3. ? (difference) A-B
4. ?? (Cartesian product) AB
5. ???? (power set)

6
? (union)

• A?B
• xx?A???B?????
• ?
• A 1, 2
• B 2, 3
• A?B 1, 2, 3

7
???? (intersection)

• AnB
• xx?A????B??
• ?
• A 1, 2
• B 2, 3
• AnB 2

8
? (difference)

• A-B
• xx?A???B??????
• ?
• A 1, 2
• B 2, 3
• A-B 1

9
?? (Cartesian product)

• AB
• (x, y)x?A???y?B??
• ?
• A 1, 2
• B 2, 3
• A-B
• (1, 2), (1, 3), (2, 2), (2, 3)
• A?n??B?m?????
• AB?nm?

10
???? (power set)

• BB?A
• ?
• A 1, 2
• ?, 1, 2, 1, 2
• A?n????? ? ?

11
?? (projection?)

• ????
• ??f f(x)y, f(U)V

f
x
y
f
U
V
A
B
12
?? (cardinality)

• ?????
• ??S1?S2?????S1??S2???1?1??????
• ????
• S1?S2????????
• ????
• S1?S2?????????????

13
??????????(?????)

• ?
• S1????
• S2????

1?1???f(2i)i
S1 -8 -2 0 2 2i 8
? ? ? ? ? ?
S2 -8 -1 0 1 i 8
14
??????????(?????)

• ?
• S1????
• S2????
• ??
• ?????(diagonalization)
• ??????????????P
• P???????
• ??????????
• 2???P???????????

1?1???f(2i)i
15
???(p. 8)

• S1(????)?S2 (????)?1?1?????????
• ??????????????
• ??i1, 2, 3, ??????i?????(1???????i???????
  ???)??????i????????10????5?????????????i??????????
  ?
• ???1??????

16
x1 x10 . x11 x12 x1i x18
x2 x20 . x21 x22 x2i x28
x3 x30 . x31 x32 x3i x38

xi xi0 . xi1 xi2 xii xi8

x8 x80 . x81 x82 x8i x88

y y0 . x11 0.5 x22 0.05 xii 510 x88 510
-i
-8
17
????

• ???? (countably many)
• ??(countable)
• ????1?1??????????
• ?
• ? ??????
• ? ???????????S??(????)??????S
• S?????
• ???? 0, 1 ???????

?????????????
18
??

• (2?)?? (binary relation)
• ??????
• ??? (??????(pair, ?))
• (1, 2) ? (2, 1)

R?AB
?? (range)
??? (domain)
A
B
19
S?????

• S????? (relation on S)
• ???????????S????????
• aRb
• (a, b)???R????

1R3 2R2 2R3
20
?????

1. ??? (reflexive)
2. ???? (irreflexive)
3. ??? (transitive)
4. ??? (symmetric)
5. ???? (asymmetric)

21
???

• ??? (reflexive)
• S???a????aRa

R
1R1 2R2 3R3
22
????

• ???? (irreflexive)
• S???a????aRa???????

R
1R1 2R2 3R3



23
???

• ??? (transitive)
• aRb??bRc?????aRc

R
1R2 2R3 1R3
24
???

• ??? (symmetric)
• aRb?????bRa

R
1R22R1 1R33R1
25
????

• ???? (asymmetric)
• aRb???bRa??????????

R
1R22R1 1R33R1



??????????????
26
???????(?1.3 p. 9)

• ????????lt
• ???
• altb??bltc???altc
• ????
• altb???blta?????????
• ?????????????

27
????

• ???? (equivalence relation)
• ??????????????????

R
1R12R23R3
???
???
???
2R33R2
28
???

• ??? (equivalence class)
• ???????????Si
• Si??, ??i?j???SinSj?
• Si???a, b????aRb
• i?j???Si???a?Sj???b????aRb???????
• S?Si???Si???????

29
??????(?1.4 pp. 910)

• ?m?????? (congruence module m)
• i-j?m????????
• im j ??? ij mod m
• ??? ???a????a-a?m??????
• ??? am b??bm c????am c
• amxb, b myc ? am(xy)c
• ??? am b???bm a
• a-bmx ? b-am(-x)

30
?????m??????
-1 0 1 2
0 -m, 0, m, 2m,
1 -m1, 1, m1, 2m1,

m-1 -1, m-1, 2m-1, 3m-1,
m?
31
?????

• R ?G ?? (G - closure)
• G?????????????????
• ????R???????????
• ??G????????????????R'

R'
R
32
?????

• ????? (transitive closure)
• R??????????R
• (a, b)?R??????(a, b)?R??
• (a, b)?R???(b, c)?R?????(a, c)?R??
• 1?2?????????R?????

1?2?3 ? 1?3
33
??????????

• R???????????R
• R? (a, a)a?S

R
R
R
34
???(1.2? pp. 25)

• ???????????
• ?????
• ???
• ??????????????
• ???????

35
??????

• ??? G(V, E)
• V ??????(vertex, node)???
• E ????((v1, v2) ???)??????(edge)???
• ? (?1.1 p.3)
• V1, 2, 3, 4, 5
• E(n, m)nm4 ????nm7

36
?

• ? (path)??
• ?????????? v1, v2, , vk (k?1)???????????
• (v1, v2), (v2, v3), , (vk-1, vk)??????????????
• ?? (cycle) vivk???
• ???(?1.1 p.3)
• 1, 3, 4
• 2
• 2, 5

37
?????

• ????? (directed graph, digraph) G
• G(V, E)
• E???????(arc)
• v?w v??w???????
• ? (?1.2 p.3)
• ( 1, 2, 3, 4 , i?jiltj )

?? (successor)
?? (predecessor)
38
???????

• ??????? (path)
• vi?vi1 (1?iltk) ?????????????? v1, v2, , vk
  (????k?1)
• ? (?1.2 p.3)
• 1?2?3?4 1??4???

39
?

• ? (tree, ordered directed tree)
• ????????????
• ????????????????????? (root)????????????
• ????????????????????
• ????????????????????????

40
?????

• ?????????????????
• ??????????????
• ???(?????)?????????????

? (root)
??? (arc)
41
??????

• ? (parent, father?) ??
• ? (child, son?) ??
• ? (leaf) ????????
• ?? (interior) ?? ??????
• ?? (ancestor) ??? (descendant)
• ??v1??v2????????(v1???v2??)
• ???????????????

????
?
?
?
42
???(?1.3 p.4)
??? ?????
?????
The quick brown fox jumped over the lazy dog
43
???(1.3? p. 56)

• ???????
• ??
• ??(basis)
• P(0)???
• ???????
• P(n-1)???????P(n)????????

??????
44
?????(?1.1 p. 5)

• ???????
• ??
• P(0)

45
???????(??1)

• ???????
• ??????
• ????n????????????

?????
46
???????(??2)

• ???????

??
47
?????

• ??
• ????
• (?)a, b, c, , 1, 2,
• ??? (string)?(word)
• ?????????????
• (?)abc, cba, a1, 2c
• w
• ???w??? (length)
• (?)abcb???abcb4
• ??e
• ???0(e0)????

48
??????

• ???(prefix)
• ???(w)??????(???0w)
• (?)abc????e, a, ab, abc
• ???(postfix)
• ???(w)??????(???0w)
• (?)abc????e, c, bc, abc

??(proper)???/???
49
??????

• ??(concatenation)
• 2???????????
• (?)dog?house???doghouse
• ????
• ??
• ???w?x???wx
• ???e
• ewwew

50
??????????

• ???????(alphabet)
• ????????????
• (?)q, z, 1 0
• () ?????????????
• ??(language, formal language)
• ?????????????????????
• (?) ????e

51
??

• ? ???????0,1????(palindrome)
• ??????
• e, 0, 1, 00, 11, 010, 11011,
• ?????????????????
• ???????(?????)?????
• ? ???????S???????????S
• Sa????Se, a, aa, aaa,
• S0, 1????Se, 0, 1, 00, 01, 10, 11, 000,