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אאא المؤسسة العامة للتعليم الفني والتدريب المهني
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א א א א א אאאאאאאאאא א א א א א א א
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• אאאאאK • אאאאאK • אאאאאאK • אאאאK
אאאאאאW אאאאאאW
אאאאאא אאאאאאאאאאא
אא א א א א א א אאאאאKK
אאאאאאאאאאKK א א אא אאא א
K אאאאאW
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1
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א 115 אא
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אא
אאWאאאאאאאאאאאKKKK
אאWאאאW
o אK0 o אאאאK o אאאK o אאאאאK o אאאאאאK o אאאאאאK o אאאאאאK
אאWK
א 115 אא
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אאWא 1Kא
אאאאאKאאW
J אא10,8,6,4,2K J אאאK J אאאK J אאאאK
אא،אא
אKאKאאK
2.1Kא FEWYXBA ,,,KKKא
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אAאπ,1,0,2−W},1,0,2{ π−=A
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3.1Kא אW
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AאאאK אאאאאK
א 115 אא
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• אאW אKAאאאא19W
אא
אאאKאאאאWאאאאא}2,8,6,4,4,2{=A
• אאW אאאKא
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}82,,:{ ≤≤∈= xزوجيxNxxANאאאK Aאאאxx2
8K
4.1Kאא BאAאאA
BאAאWAB ⊂ W
AxBxAB ∈⇒∈∀⇔⊂F∀WE
אAB ⊆BA ≠BAAB ⊂KאBAAB ⊄K
1WאאW}12,11,3{}24,5{}24,11,5,3{ === CBA אאBCAWAB ⊂AC ⊄א12
A
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= }8, ,2,6,4{A
א 115 אא
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5.1K א AB BA =אא
AxBxوBxAxABوBABA ∈⇒∈∀∈⇒∈∀⇔⊆⊆⇔=
2WאאW}0,:{}1,0{ 2 =−∈== xxNxxBA אW
אAאBאאאW
1,0100)1(2 ==−=⇒=−=− xxأوxxxxx W}1,0{=BBAא =
6.1Kאאאא אא
אאאKאאאאאאK
אאאUKא}21,7,2,5{−=U}21,2{=A}21,7,5{−=BאAB
UK אאאאφ{}Kא
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K
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א 115 אא
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1Wאא
J אאאאK J אאK J אאאאא5 J אאאא12K J אאאK
2Wאאא
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3Wאא W
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=−−==−−−−−=
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4WאאW }9,8,....,2,1{,}9,7,5,3,1{,}5,4,3,2,1{,}5,3,1{,}3,1{,}1{, ====== UEDCBAφ
אאאאW
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∉∈∈⊄⊆∈∈=
φφφ
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2Kאא 1.2Kא
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א 115 אא
א אא
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א
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5אאאא6אאK
3.2KאאאFאE CBA ,,W
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4.2Kא אBאAאאA
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א 115 אא
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אא
UA B
BA − 5WאW}20,18,12,6,3{},20,12,6,5,1{ == BAאW}5,1{=− BA
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5.2Kא אUA،Aאא
UאA FAU −KEAAW }:{ AxوUxxAUA ∉∈=−=
א
U
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א .)5,)4,)3,)2,)1 AAUUAAUAA ===== φφφIU
א 115 אא
א אא
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6.2K ABאאUאW
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BABAbBABAa UIIU == )) אא
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א U
A B
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BABAABBABAABBA
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4Kאא אאאא
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א 115 אא
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אאא אאאNא0WK
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UA B
C
א 115 אא
א אא
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7WאאאK 8WWABABA =)()( UIU 9WאBABA I=−W
)()()()( ABBABABA −−=− UIU 10WאאאאW
2)5)424)30)22)1 eπ−
א 115 אא
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1KאאW
אאW0, 1, 2, 3, 4, 5, 6, 7, 8, 9 אא0 9 وאא،אא،
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אאאK אאאאKא
אאאאW L)10(10000)10(1000)10(100)10(10)10(1 43210
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2W
)103()106()105()104()102()101(563.124
)103()106()105()001.03()01.06()1.05(563.0
)101()102()104()1001()102()14(124
321012
321
210
−−−
−−−
×+×+×+×+×+×=
×+×+×=×+×+×=
×+×+×=×+×+×=
אאאאאאא10F10E
א 115 אא
א אא
- 14 -
2KאאW אאאא0א2K،אא
2F2KE אאאאאאW
L)2(16)2(8)2(4)2(2)2(1 43210 אאאאאW
L)2(015625.0)2(0625.0)2(125.0)2(25.0)2(5.0 54321 −−−−− 3Wאאא015אאK אW
אא אא 0 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001
10 1010 11 1011 12 1100 13 1101 14 1110 15 1111
אאאאאא،אאאאאא1111
4WאאאאאאאW 111.10)31011.0)21101101)1
א 115 אא
א אא
- 15 -
אW
6875.20625.0125.05.002)21()21()21()20()21(111.10)3
6875.00625.0125.005.0)21()21()20()21(1011.0)2
109643208401)21()21()20()21()21()20()21(1101101)1
32101
4321
6543210
=++++=×+×+×+×+×=
=+++=×+×+×+×=
=++++++=×+×+×+×+×+×+×=
−−−
−−−−
אאאאאW 1Eא2אאאאאK 2Eאא2אאאאאK 3Eא0K
5WאאאאאאאW 45)319)212)1
אW 1E2אאאW
62
12=א0
326=א0
123=א1
021=א1
12אאW1100 2E2אאאW
92
19=א1
429=א1
224=א0
122=א0
א 115 אא
א אא
- 16 -
021=א1
19אאW10011 3E2אאאW
22245
=א1
112
22=א0
52
11=א1
225=א1
122=א0
021=א1
45אאW101101K Wאאאאא אאW،
W 1Eא2אאאא،
אאא،KK 2Eאאאא2،
،אאאאאאאK 3Eאא0K
6WאאאאאאאW 3125.12)2625.0)1
אW 1E2אאאאאW
25.12625.0 =×אא1 5.0225.0 =×אא0
א 115 אא
א אא
- 17 -
0.125.0 =×אא1 אא0K
א0.625אאW0.0101K 2EאWא12אאא،0.3125K
א1אאא12אא1100K א0.3125W2אאאא
Wא 625.023125.0 =×אא0 25.12625.0 =×אא1 5.0225.0 =×אא0
0.125.0 =×אא1 אא0K
א0.3125אאW0.0101K 12.3125אאW100.0101K
3KאאאאW
אאW אאאאאאאאאאאא
אאW
111111001101001
1011110000
=++=++=++
=+=+=+
7WאאאאW 100110)411111)310100)21111)1 ++++
א 115 אא
א אא
- 18 -
Wא 1E1101111 =+W
011
1111
11
+
2E11010100 =+W
011
01001
00
+
3E101011111 =+W
0101
11111
111
+
4E1010100110 =+W
0101
001011
001
+
אאW אא،אאאאא
אאאאאK 8WאאאאאאW
010101.0,10000,01.1010,1101 אW
010101.001.1010110110000 >>>
א 115 אא
א אא
- 19 -
אאאאאאאאאאאW
1110101011000
=−=−=−=−
9WאאאאW 11101)21011)1 −−
אW 1E 11011 =−W
10
0111
−
−
2E 1011101 =−W
010
11101
1
−
−
אאW אאאאאאאאאאאא
אאW
111001010000
=×=×=×=×
10WאאאאW 10111001)3111101)21111)1 ×××
א 115 אא
א אא
- 20 -
אW 1E10011111 =×W
1001
1111
1111
+
×
2E 100011111101 =×W
110001
101101
101
111101
+
×
3E 110001110111001 =×W
1100011
10010000
10011001
11011001
×
אאW Kאאאאאא
11WאאאאW 10110)211110)1 ÷÷
א 115 אא
א אא
- 21 -
אW 1E1011110 =÷W
00011
1011011
2E1110110 =÷W
0010
01010
1111010
4KאאאאאאW 1Wאnאאאאאא012 −nK
12Wא8אאאאאא0255K א16אאאא065535K
אאא؟ Kאאאא
אא JW ،אאאאW0א1،א
אאאאאK 13Wאאאאאאא Jא8W
19)219)1 − אW
1Eא2א5א19אאW10011K אא Jא8W00010011K
2E19−אא JW10010011K
א 115 אא
א אא
- 22 -
14Wאאאאא JאאW 00101)210010)1
אW
5)22(00101)2
2)2(10010)102
1
=++=
−=−=
2Wאnאא Jאאאא12 1 +− −n12 1 −−nK
15Wא8אא Jאאאא127−127K
א16אאאא32767−32767K א1אW
אאאאאאא Jאאא Kא01אאא،אא
16Wאאאאאאא1א8W 19)219)1 −
אW 1Eא1א13א19אא Jא8
W00010011אאא1K 2Eא19−אא1،000100111
W11101100K 17Wאאאאא1אאW
11101000)200010111)1 אW
23)12416()2222()00010111(11101000)2
2316421222200010111)10124
4210
−=+++−=+++−=−=
=+++=+++=
3Wאnאא1אאאא12 1 +− −n12 1 −−nK
א 115 אא
א אא
- 23 -
18Wא8אא1אאאא127−127K
א16אאאא32767−32767K א2אW
אאאאאאא Jאאא،אאאאא01א 1א
אK 19Wאאאאאאא2א8W
19)219)1 − אW
1Eא1א13א19אא Jא8W00010011אאא2K
2Eא19−אא2،000100111W111011001W11101101
אאאאFא8WE 76543210 2,2,2,2,2,2,2,2 −
20Wאאאאא2אאW 10101010)201010110)1
אW
861283282222210101010)2
86641642222201010110)17531
6421
−=−++=−++=
=+++=+++=
4Wאnאא2אאאא12 −− n12 1 −−nK
21Wא8אא2אאאא128−127K
א16אאאא32768−32767K
א 115 אא
א אא
- 24 -
אאאאאאאא2אאאאאא
אא1אW 11111111,00000000
א JW 10000000,00000000
5KאאW אאאאW
FEDCBA ,,,,,,9,8,7,6,5,4,3,2,1,0 אא16F16KE
אWאאאאאאK 22Wאאאאאאאאא
אW א א א
0 0000 0 1 0001 1 2 0010 2 3 0011 3 4 0100 4 5 0101 5 6 0110 6 7 0111 7 8 1000 8 9 1001 9
10 1010 A 11 1011 B 12 1100 C 13 1101 D 14 1110 E 15 1111 F
א 115 אא
א אא
- 25 -
אאאאאW 1Eאאא4K 2Eא4אאאK 3EאאאאאK
23WאאאאאאאW 011010011111110001)20101111100101001)1
אW
16
16
169310010110000111110011100101100001111111011010011111110001)2
5701110101101011000101111100101001)1
F
CA
===
==
אאאאאW 1Eאאא4K 2EאאK
24WאאאאאאאW
161616 9742)38)2410)1 ECFA אW
000010100101110100100100011110019742)3001110110011111011101000111111008)2
10010000101000100101000000001410)1
16
16
16
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ECFA
אאאאאאאאאאאאK
25WאאאאאאאW
1616 85)21)1 AC אW
26932222201101010000101011000101085)2
2822211100110000011)1119720
16
43216
=++++===
=++===
A
C
W
269351282560)165()168()1610(85)2
281216)1612()161(1)1012
16
0116
=++=×+×+×=
=+=×+×=
A
C
א 115 אא
א אא
- 26 -
26WאאאאאאW 650 אW
16אאאW
4016650
=א10=A
21640
=א8
0162=א2
650אאW1628A
1Wא6אאאאW 564.13)4920,671671920)3692,5454692)21386)1 ==
2Wאאאאא10W 001.0)5000,000,1)4000,10)3100)210)1
3Wא4؟ 4WאאאאW
1100)61001)51000)4111)3100)211)1 5WאאאאW
1010.1011010)4111.1000001)301.101010)211.110011)1 6WאאאW
1E؟ 2E3؟ 3E4؟ 4E10؟ 5E11؟ 7W؟אאאאאא
205)681)568)449)335)217)1 8WאאאאW
186)6125)548)424)317)210)1
א 115 אא
א אא
- 27 -
9WאאאאW 34.12)40981.0)3246.0)232.0)1
10WאאאW 10111101)5110111)411101)31010)2111)1 +++++
11WאאאW 1011111010)5111110)4101110)3100101)2111)1 −−−−−
12WאאאW 11011110)51101001)4101111)310100)21111)1 ×××××
13WאאאW 1001100)3111001)210100)1 ÷÷÷
14Wאאאאא8FאאאEאאאאW
57)699)5123)4100)385)229)1 −−−− 15WאאאאאאאW
01110100)410111111)301110100)210011001)1 16WאאאאW
16161616 98)485)314)259)1 DACA 17WאאאאW
1111110000)410111)310)21110)1 18WאאאאW
16161616 98)485)314)259)1 DACA 19WאאאאW
6500)433)314)28)1
אאאאאאאא
אא
א
א
א
א
2
א 115 אא
א אאאא
- 28 -
אאאא
אאWאאאאאאאKKK
אאWאאאW
o אאאאאK o אאאאאK o אאאאאK o אאאאאאK o אאאאK
אאW12K
א 115 אא
א אאאא
- 29 -
אאאא 1K
אאאאאKאK
אאאאאאאאFאאEאK
אאאאKאאאאא):( TTrue):( FFalseK
אאK 2Kאאאאא
אאאKאאW
1W aE אא bE אאאאo180 cE 124 =÷ dE 3=xW162 =x eE ؟א fE אK
،אאאאאאאאKאאא
א،K אאאא
אKאאאאK
א 115 אא
א אאאא
- 30 -
2WאאW aE bE אABCDcm4 cE אא
אא،אאאאאאאאאאא?W??K?אא?Wא
ABCD??cm4?אא?Wא??אK? אאאאאאאא
אאאK אאאאאא
אאא.....,,, srqpאאאאK 3Kאאא
אאאאאאW • אא)( nconjunctio?W?""andW∧ • אא)( ndisjunctio?W?""orW∨ • אא)(negation?W?KKKK""not W¬ 1.3Kאqp ∧
אאאא?F?""andKEאאWqp ∧
?pqK?אqp ∧אאאpqKאqp ∧אpq،א
qp ∧אאKאqp ∧אאא1K
אאאאpqqp ∧،אאאאpqqp ∧אK
א 115 אא
א אאאא
- 31 -
p q qp ∧T T T T F F F T F F F F
1א אאא)(Tא)(Fאא
אpq،qp ∧אאpqK
3WאאאאW 1E אאא 2E א724א =+ 3E א6א2 4E א60
אאאאאא،אאאאאאאאK
2.3Kאqp ∨ אאא
א?F?""orKEאאאWqp ∨ ?pqK?אqp ∨אאאpqK
אqp ∨אאאאpq،qp ∨אpqאK
אqp ∨אאא2K אאqp ∨אאpq
אK
א 115 אא
א אאאא
- 32 -
p q qp ∨T T T T F T F T T F F F
א2 4WאאאאW
1E אאא 2E א724א =+ 3E א6א2 4E א60
אאאאאאאאאאאאK
??אKא?p
q?pq،אאK??א?pq?pq،
אאKא?1אL2Lא??אFאKE
אא??אאאKqp ∨א?p|qK?
3.3Kאp¬ אאKא
??KKKKKKKאאאא،אpWp¬K pKp¬pא،pp¬א
pp¬אW
א 115 אא
א אאאא
- 33 -
5WאאאW aE א
bE אא
cE א
אא?Wא??אK?אא?Wאא??אאK?אאW
?א?א?Wא?אK
אאאאpq،
אאאאאאאאKא
rqp ,,אאאW
p q rT T TT T FT F TT F FF T TF T FF F TF F F
p p¬T FF T
א 115 אא
א אאאא
- 34 -
4.3Kאאא ...),,( rqpPאאאאא...,, rqp
FאאEאאא¬∨∧ ,,FKEאאאאאא،אא
אאKאאאאאאאאאאאאW
6WאאאW)( qp ¬¬ ∧ אW
אW אאאpq،
אאא)(T)(FאאK
אאאאאאאאKא
אאאאאא¬∨∧ ,,אKאאאאאאK
אא)( qp ¬¬ ∧אpqאאא،אאאאKא
אאW
אאאאאאW
א)(¬א)(∧אא)(∨K qp ∧¬qp ∧¬ )()( qp ∧¬
p q q¬ qp ¬∧ )( qp ¬¬ ∧T T F F T T F T T F F T F F T F F T F T
p q )( qp ¬¬ ∧T T T T T F F T T F F T
א 115 אא
א אאאא
- 35 -
5.3Kאאא א)(Tאאא،FאE
אא،אאאאאK אא)(FFאא)(FE
אK 7WW)()() qpqpb ∨∧∧ ¬א)() qppa ∨∨¬
אW a(אאW
)( qpp ∨∨¬א)(Tא،אW
Tqpp ≡∨∨¬ )( F≡אאE
b(אאW W
)()( qpqp ∨∧∧ ¬ )(F،א،W
Fqpqp ≡∨∧∧ ¬ )()(
6.3Kאא FEאא،
אאW...),(...),( qpQqpP ≡F≡אKE 8WאאאאWqpqp ¬¬¬ ∨≡∧ )( אWאW
p q p¬ qp ∨ )( qpp ∨∨¬T T F T T T F F T T F T T T T F F T F T
p q qp ∧ qp ∨ )( qp ∨¬ )()( qpqp ∨∧∧ ¬T T T T F F T F F T F F F T F T F F F F F F T F
א 115 אא
א אאאא
- 36 -
אא
)( qp ∧¬qp ¬¬ ∨W qpqp ¬¬¬ ∨≡∧ )(
7.3Kאא אאאאאrqp ,,אאאאא،
אאאK
p q qp ∧ )( qp ∧¬ p q p¬ q¬ qp ¬¬ ∨T T T F T T F F F T F F T T F F T T F T F T F T T F T F F F T F F T T T
1Eאאא
pqqpb
pqqpa
∧≡∧
∨≡∨
)
)
5Eאא
pTpdFFpc
pFpbTTpa
≡∧≡∧
≡∨≡∨
))
))
2Eאא
rqprqpb
rqprqpa
∧∧≡∧∧
∨∨≡∨∨
)()()
)()()
6Eאא
TFcFTc
FppbTppa
≡≡
≡∧≡∨
¬¬
¬¬
))
))
3Eאא
)()()()
)()()()
rpqprqpb
rpqprqpa
∧∨∧≡∨∧
∨∧∨≡∧∨
7Eא pp ≡¬¬
4Eאא
pppb
pppa
≡∧
≡∨
)
)
8Eא
qpqpb
qpqpa
¬¬¬
¬¬¬
∨≡∧
∧≡∨
)()
)()
א 115 אא
א אאאא
- 37 -
8W א)2( aFאEאאאא
K
rqp ∨∨ )(אאא)( rqp ∨∨אא،W
)()( rqprqp ∨∨=∨∨
9WאאWppqp ≡∨∧ )(FאE
אW אאFpEא
Fpqp ∨∧ )(Eא،Wppqp ≡∨∧ )( אאאKא
Wppqp ≡∧∨ )( אאאאא
אאKאאאאK
10WאאאאW Tqqpqpbpqpqpa ≡∨∨∧∨≡∧∨∨ ¬¬¬¬¬¬ )}(){())()()
p q r qp ∨ rqp ∨∨ )( rq ∨ )( rqp ∨∨
T T T T T T T T T F T T T T T F T T T T T T F F T T F T F T T T T T T F T F T T T T F F T F T T T F F F F F F F
p q qp ∧ pqp ∨∧ )(T T T T T F F T F T F F F F F F
א 115 אא
א אאאא
- 38 -
אW a(
b(
אאאאאאאאאאא
12WאW)1()()()( qprqpqp ¬¬¬¬¬ ∨∨∧∧∨∧ אW
אא א )()()()( qpqpqpqp ∧∨∧≡∧∨∨ ¬¬¬¬¬ Fa8E
)( qqp ∨∧≡ ¬¬ אFb3E Tp ∧≡¬ אFa6E
p¬≡ אFd5E
אא א qppqqqpqp ∨∧∨≡∨∨∧∨ ¬¬¬¬¬ )}({)}(){( אFa3E
qFq ∨∨≡ ¬ }{ אFb6E qq ∨≡¬ אFb5E
T≡ אFa6E
אא א )()]([)()1( qprqpqp ∧∨∧∧∨∧≡ ¬¬¬¬ Fa8E
)])(([)( qrqpqp ∨∧∧∨∧≡ ¬¬¬ אFb3E )()( qpqp ∧∨∧≡ ¬¬¬ אF9E )( qqp ∨∧≡ ¬¬ אFb3E
Tp ∧≡¬ אFa6E
p¬≡ א Fd5E
א 115 אא
א אאאא
- 39 -
8.3Kאאאא a(אא
אא?אpq?)( qthenpifpq،אWqp →Kאאqp →
?pq??pאqK? אאW
אqp →אאqp،אאKאא
אא،אאאאאאKאא
?אpq?pאאpqאאא،
W 13WאאאWאאאWqאWp
qp →WאאאאאאppqK
אאאאאאאאW
14WאאאW853 =+Wq83 =Wp qp →Wא83 =853 =+
qp →אאqpKpqאqpK
15WאאאW324 =÷WqאWp qp →Wאא324 =÷
pqאאאאqp →אK
אאאאא)(→אא)(∨Wqpqp ∨≡→ ¬
p q qp →T T T T F F F T T F F T
א 115 אא
א אאאא
- 40 -
א א א אאW
א א א א Wqpqp ∨≡→ ¬
?אpq??pqK? 16WאאאW
pqqppqqp ¬¬¬¬ →→→→ ,,, אW
אאW
qppq
pqqp¬¬
¬¬
→≡→
→≡→
W
pq →qp ¬¬ →pq ¬¬ →אאאqp →
אאאqp →אpq →qp ¬¬ →K
17WW4||: <xq4: 2 =xpKאW qp →Wא42 =xF2±=xE4|| <xאK pq →Wא4|| <x42 =x،א1=x3−=x qp ¬¬ →Wא42 ≠x4|| ≥x،א1=x3−=x
אאאW
Wqpqp ∨≡→ ¬אאW
qpqpqp ¬¬¬¬ ∧=∨≡→ )()(
p q qp → p¬ qp ∨¬T T T F T T F F F F F T T T T F F T T T
p q qp → pq → p¬ q¬ qp ¬¬ → pq ¬¬ →T T T T F F T T T F F T F T T F F T T F T F F T F F T T T T T T
p q qp → )( qp →¬T T T F T F F T F T T F F F T F
א 115 אא
א אאאא
- 41 -
b(אאא אאאא?pא
q?Wqpא ↔K אאאpq
אKאאאאW אאאW
)()( pqqpqp →∧→≡↔ FאאאE
9.3Kא
אאאnPPP ,...., 21אאאQ،אW
QPPP n a,...., 21 אQPPP n a,...., 21FEאאFQE
אnPPP ,...., 21،FEאאK
18WאאW FEpqqpb a,) →FEqqppa a→,)
אW a(pqp →א
אqאאאKאאאאW
אאאא،אאא
אqאאאאאKאWqqpp a→,K
p q qp ↔T T T T F F F T F F F T
p q qp →T T T T F F F T T F F T
א 115 אא
א אאאא
- 42 -
b(אאאa(אW אאqp →qאאא
אא،אpאאאאאFpEא،pqqp a,→FKE
p q qp →T T T T F F F T T F F T
אאאnPPP ,...., 21א
אnPPP ∧∧∧ .....21אFאEא،QPPP n a,...., 21FEאאQ nPPP ∧∧∧ .....21،W
TQPPP n ≡∧∧∧ )....( 21 aFאKEאאאW 19Wאrprqqp →→→ a,
אW א)()()( rprqqp →→∧→ aאאא
W
Trprqqp ≡→→→∧→ )()()(א)()()( rprqqp →→∧→ aFKE
p q r qp → rq → rp → )()( rqqp →∧→ )()()( rprqqp →→→∧→
T T T T T T T T T T F T F F F T T F T F T T F T T F F F T F F T F T T T T T T T F T F T F T F T F F T T T T T T F F F T T T T T
א 115 אא
א אאאא
- 43 -
20W؟אא אאאאKא
אא،אKאK אWאאאW
אW1p¬אW1p אאW2p¬אאW2p
W221 , ppp ¬→אW1p¬ אW1221 , pppp ¬¬→ aאאWTpppp ≡∧→ ¬¬
1221 )( a
אאTpppp ≡∧→ ¬¬1221 )( aאK
10.3Kאא א,...),( qpPא,...),( qpQא,...),( qpQ
,...),( qpP،W ,...),(,...),( qpQqpP ⇒
21Wpqp ∨qp ∨א pאאqp ∨،
אאאKאqpp ∨⇒K אא,...),(,...),( qpQqpP a
،,...),( qpQ,...),( qpPאKאQP aא
אאאQP →אאKאאאאW
1p 2p 1p¬ 2p¬ 21 pp → 221 )( ppp ¬∧→ 1221 )( pppp ¬¬ →∧→T T F F T F T T F F T F F T F T T T T F T F F T T T T T
p q qp ∨T T T T F T F T T F F F
א 115 אא
א אאאא
- 44 -
,...),( qpP,...),( qpQאאW • ,...),( qpP,...),( qpQ • א,...),(,...),( qpQqpP aFE • אאQP →אFאE
4KאאאאאW אאאאאאW
א א א א א א
אא אא אא אא אא א
אא אא אא אא
1WאאאW dE א eE אאאא fE ؟ gE א522 =+אא
aE אbE א cE dE 5845 >
2Wאאא?WאW?q?אW?p Wpqdqpcqpbpaא ¬¬ ∨∨∧ ))))
3WאאאW ?אW?r?W?q?אW?p
א 115 אא
א אאאא
- 45 -
∧∨¬אאאאאאא ,,W aE אא bE אאא cE אא dE אא 4WאאאW
א:sא:r:qא:p אאאW
srsqrqsprpqpbsrsqrqsprpqpa ∧∧∧∧∧∧∨∨∨∨∨∨ ,,,,,),,,,,) 5WאאWFqpqp ≡∨∧∧ ¬ )()( 6WאאאאאאW
qqppcqqpqpbqppa ∨∨∧∨∧∧∨∨∨ ¬¬¬¬¬¬ )}({))}(){())() 7WאאאאW
qpqpcpqppbrpqprqpa ¬¬¬ ∨≡∧≡∧∨∨∧∨≡∧∨ )())())()()() 8WאאאאW
qqpqqppcTqqppbTqppa ≡∧∧∨∨∧≡∨∨∧≡∨∨ ¬¬¬¬¬ )}({)}({))}({))() TqpqpqpqpeTqqpqpd ≡∧∧∧∨∧∨∧≡∨∨∧∨ ¬¬¬¬¬¬¬ )()()()())}()){
9WאW)()()())}({))()() qprqpqpcqpqpbrqpqpa ¬¬¬¬¬¬¬¬¬¬¬¬¬ ∨∨∧∧∨∧∨∧∧∧∧∨∧
10WאאאאאאאW a(אאא b(אא،אאא
א c(אאאא
11WאאאW 731 (10113א+= =+b431 (1183א+= =+a
431 (1083א+= =+d731 (1183א+= =+c
א 115 אא
א אאאא
- 46 -
12WאאאW אABCאא:qאABCא:p
،qpאאאא → 13Wאאאאאאא
אאא)bאאא)a FאWqpqp ∨≡→ ¬E
14WאאאW a(אא b(אאאא c(אא
15WW
TpqqpfTrprqqpe
TqpqpdTqpqpc
TqqppbTppa
≡→↔→≡→→→∧→
≡∧↔∨≡∨↔∧
≡→→∧≡↔
¬¬
¬¬¬¬¬¬
¬¬
)()())()]()[()
)()())()()
)]([))
16WאאW )()())()())()() rqprqpcqpqpqpbpqqpa ↔↔≡↔↔∧∨∧≡↔→≡→ ¬¬¬¬
17WW pqqpb ¬¬→ a,)qpqpa ¬¬→ a,)
prqrqpc ¬¬ →→ a,,) 18WאאW
אא،אאאK 19WאאW
7Kא747K74א 20WאאW
)()()5)4)3)2)1 qpqppqqpqpp ∨∧∧∨∨∧ ¬¬¬ 21WאאאW
CBACBACBABA ⊂⊕= UIUI )4)3)()2)1
אאאאאאאאאא
אא
א
א
א
א
א
3
א 115 אא
א אאאאא
- 47 -
אאאאא
KאאאאאאWאא
אאWאאאW o אאאאאK o אאאK o אאאאK o אאאאאאK o אאאאאאNANDK
אאWK
א 115 אא
א אאאאא
- 48 -
אאאאא
1K אאאאאא
KאאאאאאאאאW
aאp aאp¬ ×א∧Fא×،ba ×abE +א∨ 1אW)(T 0אW)(F
אאאאאאאאW
אאא א
121110987654321
FFppFpFppTpp
pTpTTppppppp
qpqp
qpqp
rpqprqprpqprqp
rqprqprqprqp
pqqppqqp
=∧=∨=∧=∨
=∧=∨=∧=∨
∨=∧
∧=∨
∨∧∨=∧∨∧∨∧=∨∧
∧∧=∧∧∨∨=∨∨
∧=∧∨=∨
¬¬
¬¬¬
¬¬¬
,,
,,
)(
)(
)()()()()()(
)()()()(
00,00,1
1,11,
)()()()()()(
)()()()(
=×=+=×=+
=×=+=×=+
+=×
×=+
+×+=×+×+×=+×
××=××++=++
×=×+=+
aaaaaaa
aaaaaaaaa
baba
baba
cabacbacabacba
cbacbacbacba
abbaabba
א 115 אא
א אאאאא
- 49 -
1WאאאWcbaZbcbabcaZa +=++= )))(() אW
2Kא
אאאאKאאW ?אאאאא?
W OR:+אא:bא:aא:Z
אFEW baZ +=
אאW
אאאאKאאKאאW
2WW)( cbaacab +=+ אW
)a א )b א
)(
0000
)()(
))((
bacZbcac
bcacbccaacb
bccbcbaacbaa
cbabccbaa
cbabcaZ
+=+=
+++=+⋅⋅++⋅=
+++=
+++=
++=
5
12
12
11,9
5
)(
)(
)(
cba
cba
cba
cbaZ
+=
+=
=
+=
8
7
a b baZ +=
1100
1010
1110
א 115 אא
א אאאאא
- 50 -
אאאאאאאא10TFW
a b c ab ac acab + cb + )( cba +
11110000
11001100
10101010
11000000
10100000
11100000
11101110
11100000
אא6Facab +EאאF)( cba +E،אW
)( cbaacab +=+ 3Kאאאאא
אאאFאאEאאאאאKאאאאא
LאאKLאא01אאא
א01אאKאאאFאאאאאEL
אאאKאאאאאK
4Kאאא א،אKאא
אאאאאאאאאאאK
א 115 אא
א אאאאא
- 51 -
1.4KאאANDFE אANDF1EאאאאF1KEFאa1E
אANDKאWabZ =KאFא1EאאאAANDBF1=aAND1=bKEאb1אאANDK
אאZאANDאFEabKא0=Zא0=a0=bK אcba ,,אWabcZ =
2.4KאאORFE
אאORF1EאאאאF1EKאFאa2EאORKאW
baZ +=KאF1=ZEאאF1=aOR1=bKEאb2אאANDKאאZאORאFEabK
א1=Zא0=a1=bK
3.4KאאNOTFאE
،אאאW 1=a0=Z0=a1=Z
a b abZ =
1100
1010
1000
a b abZ =
1100
1010
1000
a bאa1
a
b
Z=abAND
אb1
a
• •b
אa2
a
b
Z=a+bOR
אb2
א 115 אא
א אאאאא
- 52 -
אאZאK،אאW aZ =
אאNOTאW
4.4KאאNAND אאNANDאאANDNOTKאאאNANDאאאANDKאאאNANDאאAND،W
abZ = אאNANDאW
5.4KאאNOR
אאNORאאORNOTKאאאNORאאORKאאאNORאאOR،WbaZ +=
אאNORאW
a aZ =
10
01
a b ab abZ =
1100
1010
1000
0111
a b ba + baZ +=
1010
1100
1110
0001
a aZ =NOT
a
b
NAND abZ =
a
b
baZ +=NOR
א 115 אא
א אאאאא
- 53 -
3WאZאאאW a
Z
b•
AND NOT
NOT
OR NOT
OR
c
אW אאאANDWbandaאאאANDWab אאאNOTFאWEabאאאNOTFאWEa
אאאORWcandaאאאORWca + אאאNOTWca +אאאORWabandca +
אאZאאORWbaabZ ++= 4WאאZאאאאאאאאאאK אW
אאאNOTWa אאאANDWbanda אאאANDWba
אאאORWaandba אאאאORWbaaZ +=
אאW F6E))(( baaabaaZ ++=+= F1110EbaZ +=
אאאאORאW
ZOR
a
b
Za
b
NOTAND
OR•
א 115 אא
א אאאאא
- 54 -
5WאאאאW
אW
אאאANDFאאWEcandbandaאWabc אאאANDFאWEcandbandaאWcba
אאאANDFאWEbandaאWba אאאאORWbaandcbaandabc
אZWbacbaabcZ ++=
6WאאאאאאK
אW א1W
bba
bba
aabba
abbaba
abORbaORbaZ
+=
+=
++=
++=
=
1
)(
אאאאאאאW
a b Z
1100
1010
1011
א 115 אא
א אאאאא
- 55 -
6.4KאאאאאNAND
אאאאאאאאאאאאאNANDKאאאאאא
אאאאאאאאאK אאאאאNANDW
NOT
AND
OR
•NANDab NAND
a
bababZ ==
aa • NAND
b • NAND
NAND
b
bababaZ +=+=⋅=
a aaaZ =⋅=• NAND
א 115 אא
א אאאאא
- 56 -
אאאאאאNANDאאאאאאאNANDאKאאNANDא
אאאK 7WאאאאאאאNANDK
אW
אאANDWcandbאWbc אאאORWdandbcאWdbc +
אאANDWaandedbc ,+ אאWadeabcedbcaeZ +=+= )(
אאאאNANDאאORאאWadeabceadeabceZ +=+=אW
adeabceZ ⋅= אאאאאאNANDW
א 115 אא
א אאאאא
- 57 -
1WאאאאאאW
)()())())))(()
))(()))()(()))(())()2342 yzyyzyhcbagaaaafbbabae
babadcacbcbacaccbabcabbabaa
++++++++++
++++++++
2WאאאאאאאאאאK
baabfbabaecababcad
abbaacabaabaabaa
+==+++=+
=+=+=+
))))(()
)())())
3Wאאא،אאאKאאאאאאאNANDאאאאK
a)
OR
OR
OR
•
AND
a
b
c
d
AND
OR
ANDOR
•
•
•
a
b
c
b)
4WאאאNANDאabbaZ ++=א
אאאאאAND،ORNANDKאאאאK
5WאאNANDאcbaZ ++=אאאאK
א 115 אא
א אאאאא
- 58 -
6WאאאאW
a • NAND
b • NAND
•
•NAND
NAND
NAND
7WWbabaS +=abC =אאאW
b •
NAND •
NAND
•
NAND
NAND
NAND
a•
S
C
8WFEאאאאאאא،אאNANDK
9WאאאאאK
a b Z
1100
1010
0101
a b c Z
11101000
11010100
10110010
10010010
אאאא
אא
א
א
4
א 115 אאא
א אא
- 59 -
אאWאא
אאWאא،אאאאאאאאאאאאK
אאWאאאW
o אאאאK o אאאאK o אאאאאK o אאK o אK o אאאאK
אאWK
א 115 אאא
א אא
- 60 -
אאWאא אאאאK
Wאא،אאאאאאאאאאK
אאאא113אאאא،אאאאK
1KאW 1WfXYאאX
אאY،1x2xXYW
)()( 2121 xfxfxx =⇒= )(xfאאxK
אXאאYאא)(xfy =xאאאfאx)(xfy =אאאf)(xfYא
xY)(xfYK אאW YXf →:
،אאאאאK
2WאאfFEאאאאאא،אאאfאאאא
אאאK אאffDאfRK
3WאfgאW
)()()2
)1
xgxfDx
DD
f
gf
=⇒∈
=
Wgf =K
א 115 אאא
א אא
- 61 -
4WאאYXf →:ZYg →: אאfg oאXZW [ ])()( xfgxfg =o
1Wאfg ogf oW
2
2
2
2
)(,RR:)(,RR:)4
)(,NN:)(,NN:)3
)(,NN:1)(,NN:)2
)(,RR:1)(,RR:)1
xxggxxff
xxggxxff
xxggxxff
xxggxxff
=→=→
=→=→
=→+=→
=→+=→
אW 1ERR: →fg oW [ ] [ ] ( ) 1211)()( 22 ++=+=+== xxxxgxfgxfg o
RR: →gf oW [ ] [ ] 1)()( 22 +=== xxfxgfxgf o 2ENN: →fg oW [ ] [ ] ( ) 1211)()( 22 ++=+=+== xxxxgxfgxfg o
NN: →gf oW [ ] [ ] 1)()( 22 +=== xxfxgfxgf o 3ENN: →fg oW [ ] [ ] ( ) xxxgxfgxfg ====
2)()(o
NN: →gf oW [ ] [ ] xxxxfxgfxgf ===== 22)()(oxאK
4ERR: →fg oW [ ] [ ] ( ) xxxgxfgxfg ====2
)()(o RR: →gf oW [ ] [ ] xxxfxgfxgf ==== 22)()(ox
אK
1WאאK 5WאאאאאאK
אאW אאאאא
אאאW 1ExFאE)(xfy =אאK 2Eא),( yxאאאK
א 115 אאא
א אא
- 62 -
3EאאאאאxKKK
אאאאאK 6WאW
1EאW)()( xfxf −=−0)()( =+− xfxffDx∈fDx∈−K 2EאW)()( xfxf =−0)()( =−− xfxffDx∈fDx∈−K
אאאW 7Wאאאאאאאאאא
אאאFKאE Wאאאאא،אאאא
אאאאK אאאWאאאאאאאאאאאאאאאK
אאאW אאאWאאאאאאאאאK
1WאאאW
1)(,RR:)41)(,RR:)3
)(,NN:)2)(,NN:)133
33
−=→+=→
=→=→
xxffxxff
xxffxxff
2Wאא1אW 3Wאfg ogf oW
xxfgxxff
xxfgxxff
xxfgexff
xxfgx
xff
x
=→=→
=→+=→
=→=→
=→=→
)(,RR:)(,RR:)4
)(,RR:1)(,RR:)3
)(,RR:)(,RR:)2
)(,RR:1)(,RR:)1
2
2
א 115 אאא
א אא
- 63 -
4Wא؟
xxffexffx
xffxxff
x =→=→
=→=→
)(,RR:)4)(,RR:)3
1)(,RR:)22ln)(,RR:)1
5Wאאאא15FאאאאWE 6WאאאW
xxffexffx
xffxxff
x =→=→
=→=→
)(,RR:)4)(,RR:)3
1)(,RR:)22ln)(,RR:)1
7WאW 50log)4360log360log)316log)210log)1 21023 =
2KאאאאאW אאאאWW⎣ ⎦W⎣ ⎦ ZR: →W⎣ ⎦x
xK 2W
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 44,77,95.8,25,314.3 −=−=−=−== אאאאWW⎡ ⎤W⎡ ⎤ ZR: →W⎡ ⎤x
xK 3W
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 44,77,85.8,35,414.3 −=−=−=−== אאאWWINTWZR:INT →W)(INT x
אאxK 4W
4)4(INT,7)7(INT,8)5.8(INT,2)5(INT,3)14.3(INT −=−=−=−== אאאWWW),0[R: ∞→Wxx
K 5W
44,77,5.85.8,55,14.314.3 =−==−==
א 115 אאא
א אא
- 64 -
אאWmkK)(mod mkkmkmK
6Wא01−mW
5)8(mod3,0)3(mod39,5)8(mod371,2)7(mod26,3)8(mod3,2)11(mod35,0)5(mod25,4)7(mod25
=−=−=−=−====
7Wאאא1224K 6)24(mod2010 =+
אאאאאK
1WאW ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
1.4,27,,2,23.15)4
)1.4(INT),27(INT),(INT),2(INT),23.15(INT3)
1.4,27,,2,23.15)2
1.4,27,,2,23.15)1
−−
−−
−−
−−
π
π
π
π
2WאW
)32(mod31),11(mod31),12(mod13),24(mod13)2),32(mod31),11(mod31),12(mod13),24(mod13)1
−−−−
3Wאאאא،W 1E20؟ 2E31؟ 3E5؟
W 4E15؟ 5E21؟ 6E43؟
א 115 אאא
א אא
- 65 -
אאWאא אאאאKK
1KאאW 1WאאKאא
אאK 1Wאא5=xW
15472)( 23 −+−= xxxxf Wא)(xfאא5=xאא،W)5(fK
אW80)5( =fK 2WאאאאאאK
2WאאKאאאK אא
אKאאאאאאאאK
3WאאW אW)(xfאאax =K
אאW)(afK אא41×FE
אKאאאאאאאאK
3WאאאאK אאK
4Wא1WאאW ( ) ( ) ( )
801595152075152017525015)54()557()5552(15545752)5( 23
=−=−+=−+−=−×+××−×××=−+−=f
אאא6123א =++3K אW
א 115 אאא
א אא
- 66 -
15)4)72((15)472(15472)( 223 −+−=−+−=−+−= xxxxxxxxxxf W
80159515519155)415(155)453(155)45)710((155)45)752(()5(
=−=−×=−×+=−×+×=−×+×−=−×+×−×=f
אאא3א3K אאאKאאאאK
אאאאאאאאW
אאW3xאK אאWאא2xK אאWאאאK אאאWאאxK
אאWאאאK אאWאאאאK
אאWאאאאK אאאאא33KK
אאא3K 5WאאW
אWאabba >K אאWאאאab؟
אW 1EאאWאאאW
אאWאאK אאWאאאK 258=a60=bW אאWא258=aW258,129,86....,,6,3,2,1
א60=bW60,30,20,15,12,10,6,5,4,3,2,1K אאWאא258א=a60=bW6K
א 115 אאא
א אא
- 67 -
2EאאWאאW אאWab1rK אאWאאאאאאאK אאWאאאאא
K אאאWאאאאאאK 258=a60=bW אאW258=a60=bאW181 =rK אאW60=b181 =rאW62 =rK אאאW181 =r62 =rאW03 =rK אאWאאאאאאאW62 =rK
Wאאאא 061860258 321 =>=>=>=>= rrrba
2KאאW אאאKKאאא
WאאK ،אאאאאאא
אאאK אאאאאאא
אאKאאאK 4Wאאאאאא
אKאאWאאאאאאאK 6Wא3אאאא4K
אא؟ אW
אאא6123 =++3،אאW9K
א 115 אאא
א אא
- 68 -
אW6K 7WאאW
אW)(xfאnaK אאWא)(afK
אאא؟א؟ אW
1Eאאא2
)1(1)1( +=++−+
nnnn Ln
אWnnnnnnnn23
21
22)1( 2
2
+=++
=++K
2EאאnnאWnnn 2=+K
אאאK אאאOאW
אאאאאאKאאאK
אאאאW n2
3n 2n nn 2logn n2log n 32 125 25 15 5 3 5
310 310 210 40 10 4 10 3010 610 410 700 100 7 100 30010 910 610 410 310 10 1000
אאאאOK 8WאאW)( 2nOא،W)(nOK
אאאאאאאK אא،אאאא
K
א 115 אאא
א אא
- 69 -
3KאאאW אאW
אWDATAWnITEMW אאWITEMDATA؟א
אאWLOCITEMDATAK אאWDATAITEMאאאאא
LOCK אאWאאKאאאK
אאWא،אאnאאKאאW)(nOK
9WאאאאW 15)7,15,3,2,0( =−−= ITEMDATA
אW אאWאא1=LOCאW
150WאאאW2=LOC؛ 152−WאאאW3=LOC؛ 153WאאאW4=LOC؛ 1515WאW4=LOCK
אאW אWDATAWnITEMW
אאWITEMDATAא؟ אאWאא??W
אאWאאאא2
1+nFאnE2nFאnE
ITEMW אITEM
אITEMITEMאאא אITEMITEMאאא
א 115 אאא
א אא
- 70 -
אאWאאאאאITEM
אא2
1++ lowhigh2
lowhigh +FאEWhigh،אlowאKK
אאWאITEMK אאW)(log2 nOK
10WאאאאW 15)16,10,5,5.4,0,1,3( =−−= ITEMDATA
אW
אאWאאא1,7,42
17===
+= lowhighLOC
אW 155.4W5.415 >אאאW5,7,6 === lowhighLOC؛
אאWW 1510W1015 >אאאW7,7,7 === lowhighLOC؛ 1516W1615 <K
אאW15DATAK אאW
אWDATAWnK אאWDATAK
אאWאא،אK
אאW،1−n)1(3 −nא)1(4 −nK
،אא)2(4 −nKאאW
)(22)1(22
)1(4)1(4)2(4)1(4 22 nOnnnnnnnn =−=−=−
=++−+− L
11WאאאאW )8,11,3,5,21,12,16( −−=DATA
א 115 אאא
א אא
- 71 -
אW אאWאאאW
1,1,16 === elcurrentLevLocationcurrentMinimumcurrentMin W
LocationcurrentMin imumcurrentMin DATA 2 12 12 2 12 21 4 5− 5− 5 3− 3− 5 3− 11 5 3− 8
אאW3−W5 אאאאאאW
)8,11,16,5,21,12,3( −−=DATA אאWאאאW
2,2,12 === elcurrentLevLocationcurrentMinimumcurrentMin אאW
LocationcurrentMin imumcurrentMin DATA 2 12 21 4 5− 5− 4 5− 16 4 5− 11 4 5− 8
אאW5−W4 אאאאאאאאW
)8,11,16,12,21,5,3( −−=DATA אאW )21,16,12,11,8,5,3( −−=DATA
א 115 אאא
א אא
- 72 -
1WאאאאW
7)1,7,8,19()44)4,23,18,19,0()313)5,13,63,89,0()263)5,13,63,89,0()1
=−=======
ITEMDATAITEMDATAITEMDATAITEMDATA
2WאאאאאאK 3Wאאאאא1K 4WאאW
אא1100אאK אא1100K
אאאאW1Eא2Eא3אEאאK
1EאאאK 2E؟אאאא
5WאOאW
nnnnn
nnnnnnn
22221
22
log)12()8log)72)62)5
log1)423)31)22log)1
+++
−+−++
6W1EאnKK 2EאאK
א 115 אא
א
אא 1E،،א،אא،
1403 J1983K 2) Alfred Aho, John Hopcroft and Jeffrey Ullman, Design and Analysis of Computer Algorithms, Addison Wesley, Reading, England, 1974. 3) Gwyn Davies and Gordon Hick, Mathematics for scientific and technical students, Addison Wesley Longman, Harlow, England, 1998. 4) Micheal Garey and David Johnson, Computer and Intractability, A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979. 5) C. L. Lui, Elements of Discrete Mathematics, McGraw-Hill, New York, 1977. 6) Alexander Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons, Chichester, England, 1986. 7) Seymour Lipschutz and Marc Lipson, Discrete Mathematics, McGraw-Hill, New York, 1997. 8) Peter Tebbutt, Basic Mathematics, John Wiley & Sons, Chichester, England, 1998.
א 115 א
א
א
אאWא 1 אאWא 3
1Kא 3 א 3
א 3 אא 4
5 אאאא 5
6 2Kאא 7
א 7 7
אאא 8 א 8
א 9 10
אא 10 4Kאא 11
12 אאWא 14
1Kאא 14 2Kאא 15
אאאאא 16 אאאאא 17
א 115 א
א
3Kאאאא 18 אא 18 אא 19
אא 20 אא 21
4Kאאאאאא 22 אא J 22 אא1 23
אא2 24 5Kאא 25
אאאאא26 אאאאא26
27 אאWאאאא 29
1K 31 2Kאאאאא 31 3Kאאא 32
3K1Kא 32 3K2Kא 33 3K3Kא 34 3K4Kאאא 36 3K5Kאאא 37 3K6Kאא 37 3K7Kאא 38 3K8Kאאאא 41 3K9Kא 43 3K10Kאא 45
א 115 א
א
4Kאאאאא 46 46
אאWאאאאא 49 1K 51 2Kא 52 3Kאאאאא 53
4Kאאא 53 4K1KאאAND 54 4K2KאאOR 54 4K3KאאNOT 54 4K4KאאNAND 55 4K5KאאNOR 55 4K6KאאאאאNAND 58
60 אאאWאא 62
אאWאא 64 1Kא 64
66 2Kאאאאא 67
68 אאWאא 69
1Kאא 69 2Kאא 71
אאאOא 72 3Kאאא 73
אא 73 אא 73
א 115 א
א
אא 74 76
אא 77
