Corrections Cours 3, Transparent 21 Miki (Nicolas) Hermann (CNRS

Corrections Cours 3, Transparent 21 Miki (Nicolas) Hermann (CNRS

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ƒ-„D…}†D‡‰ˆ$ŠŒ‹†DŽ‰Š(‘†\Ž'’!“•” Ž}–—‹˜…™]–—ŒšB‹–—ˆ$ŠY›Tœ>ž–—œ NPŸ¡ –—ˆ$™e‘†D…}Ž™!–—œe‡‘†\Ž†D‹Ž}†Dˆ:¢e‘†\Ž'£e†¤‡}†DY›T…‰Š‘–—‹Ž
¢!–l–—‘„\†D‹e‹†\Ž S Ž‰œeŠŒ¥—›T‹l…}Ž¦ (S)
§`¨ S = {[(x ∧ ¬y) ≡ z]}
©>¨ S = {[(x 6≡ y) ≡ z], [(x ∨ y) ≡ z]}
ª>¨ S = {[(x 6≡ y) ≡ z], [(x ∧ y) ≡ z]}
« ¨ S = {[(x ≡ y) ≡ z], [(x ∨ y) ∧ z]}
¬ €(­]­]®¯±°&²}€
§`¨{³ –—œŽ›´¥q–—‹Ž R = [(x ∧ ¬y) ≡ z] = {000, 010, 101, 110} ¨±µ ›¶‡}†DY›T…‰Š‘–—‹
· ‹S†\Ž‰º»…†\Ž‰¸ …(Ÿ ¥—™*›T›—ŒŽ Š‘£e§ † ¥` ›T›TŒ‡Š‘£e000
· ‹Sº»†\Ž‰…(™*›—Ž –—Ÿ ˆ$™e‘†D† ˆ¹ †D ‹l›T∈…‰‡ ŠŒR¥q111† ¹ 6∈›T‡ [(x ∧ ¬y) ≡ z] ¹
· ‹Sº»†\Ž‰…(™*›—Ž ¼F–—‡‰‹ ›T‡ ¹ ¬000 = 111 6∈ R ¹
· ‹Sº»†\Ž‰…(™*›—Ž£>œ*›T±¼F¹ –—‡‰‹ 101›T‡ ∧ 110 = 100 6∈ R ¹
· ‹Sº»†\Ž‰…(™*›—Ž{¢eŠ ½‰–—‹ …‰ŠŒ¥q† ¹ ›T‡ 101 ∨ 110 = 111 6∈ R ¹
·
maj(000, 101, 110) = 100¨ 6∈ R ¹
·¿+›T‹S‡ º»†\Ž‰–—…(‹ŽÀ™*„\›—Á˜Žsœ›†D¾‹l… ‹† ¹ ›T‡ ¹ 000
+ 010 + 101 = 111 6∈ R
†D…-’]“” (S) †\Ž‰…(™!–—Œšl‹–—ˆ$ŠY›T ¨
Pol(R ) = I
©>¨{³ –—œŽ›´¥q–—‹Ž
0
0
0
0
0
0
0
0
0
R1 = [(x 6≡ y) ≡ z] = {000, 011, 101, 110}
R2 = [(x ∨ y) ≡ z] = {000, 011, 101, 111}
µ ›¶‡}†DY›T…‰Š‘–—‹ R = [(x 6≡ y) ≡ z]
· ‹S†\Ž‰º»…†\Ž‰¸ …(Ÿ ¥—™*›T›—ŒŽ Š‘£e§ † ¹¥`›T Œ›TŠ‘‡ £e† 000›T∈‡ R ¹
· ‹Sº»†\Ž‰…(™*›—Ž –—Ÿ ˆ$™e‘†Dˆ¹ †D ‹l…‰ŠŒ¥q111† 6∈›T‡ R ¹
· ‹Sº»†\Ž‰…(™*›—Ž ¼F–—‡‰‹ ›T‡ ¹ ¬000 = 111 6∈ R ¹
· ‹Sº»†\Ž‰…(™*›—Ž£>œ*›T±¼F¹ –—‡‰‹ 011›T‡ ∧ 101 = 001 6∈ R ¹
· ‹Sº»†\Ž‰…(™*›—Ž{¢eŠ ½‰–—‹ …‰ŠŒ¥q† ¹ ›T‡ 101 ∨ 110 = 111 6∈ R ¹
·
¹ maj(011, 101, 110) = 111 6∈ R ¹
§
1
1
1
1
1
1
1
·ƒ-–—†\‹ Ž‰…›¾‹††D…{‘† Ž‰š>Ž‰¨e… µ ˆ›¶† ‡}†DŽ}Y–—›Tœ…‰Ž Š‘–—Ÿ ½‰‹ › †D‹l…†\Ž‰… x + y + z = 0 ¨
R = [(x ∨ y) ≡ z]
Š‘£e† ¹ ›T)‡ =000L ∈ R ¹
· ¸§ Ÿ ¥—¥—¹›T›TŒŒPol(R
· ‹SŸ º»†\Ž‰…(Š‘£e™*†›—¹Ž ¼F›T‡–—‡‰111
· †\Ž‰…{£>œ*›T±¼F–—‡‰‹ ‹ ¹ ›T∈›T‡ ‡ R011¹ ∧ 101 = 001 6∈ R ¹ †D…
· ‹Sº»†\Ž‰…(™*›—Ž{¢eŠ ½ œe‹ ¹ …‰ŠŒ¥q† 000›T∨‡ m = m ¹ 111 ∨ m = 111 011 ∨ 101 = 111 ∈ R ¹
· ‹Sº»†\Ž‰…(™*›—Žs›¾‹† ›T‡ ¹ maj(000, 011, 101) = 001¨ 6∈ R ¹
·ƒ-–—‹
¨ ¿+›T011‡ –—+‹Ž}101
+ 111 = 001 6∈ R
¹
\
„
l
Á
œ
†D‹˜… Pol(R , R ) = I †D…s’!“•” (S) †\Ž‰…(™!–—Œšl‹–—ˆ$ŠY›T ¨
Pol(R ) = V
¹
ª>¨{µ †\Ž:™e‡}–—™e‡‰Š‘„D…}„\Ž1£e†Y› ‡}†DY›T…‰Š‘–—‹ R = [(x 6≡ y) ≡ z] ŽÀ–—‹˜…£e„c½ –—‹e‹lœ†\Ž ¨Q³ –—œŽ›´¥q–—‹Ž:Y› ‡}†DY›T…‰Š‘–—‹
ÁlœeŠ
R = [(x ∧ y) ≡ z] = {000, 010, 100, 111}
†· \Ž‰…¸ Ÿ ¥—›TŒŠ‘£e† ¹ ›T‡ 000 ∈ R ¹
· ‹S†\Ž‰º»…†\Ž‰§…(Ÿ ¥—™*›T›—ŒŽ Š‘£e† –—¹ ˆ$ ™e›T‡‘†D111
· †\Ž‰…{¼s–—‡‰‹ ›T‡ ˆ†D‹l∈…‰ŠŒ¥qR† ¹ ›T‡ ¬010 = 101 6∈ R†D… ¹
· ‹Sº»†\Ž‰…(™*›—Ž¹ £> œ*›T±000¼F–—‡‰∧‹ m ›T=‡ 000 ¹ 111 ∧ m = m 010 ∧ 100 = 000 ∈ R ¹
· ‹Sº»†\Ž‰…(™*›—Ž{¢eŠ ½‰–—‹ …‰ŠŒ¥q† ¹ ›T‡ 010 ∨ 100 = 110 6∈ R ¹
· ‹Sº»†\Ž‰…(™*›—Žs›¾‹† ›T‡ ¹ maj(010, 100, 111) = 110¨ 6∈ R ¹
·ƒ-–—‹
¨ ¿ ›T010
+ 100 + 111 = 001 6∈ R
¹
‡
—
–
‹ŽÀ„\Á˜œ†D‹l… ¹ Pol(R , R ) = I †D…s’!“•” (S) †\Ž}…(™]–—ŒšB‹–—ˆ$ŠY›T ¨
Pol(R ) = E
¹
« ¨{³ –—œŽ›´¥q–—‹Ž
1
0
2
2
2
2
2
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2
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1
2
0
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0
R4 = [(x ≡ y) ≡ z] = {001, 010, 100, 111}
R5 = [(x ∨ y) ∧ z] = {011, 101, 111}
µ ›¶‡}†DY›T…‰Š‘–—‹ R = [(x ≡ y) ≡ z]
· ‹S†\Ž‰º»…†\Ž‰§…(¥—™*›T›—ŒŽsŠ‘£e¸ † Ÿ ¥`›TŒ›TŠ‘‡ £e† ¹ ›T‡ 000 6∈ R ¹
· ‹Sº»†\Ž‰…(Ÿ ™*›—Ž –—¹ ˆ$ ™e‘†D111
· ‹Sº»†\Ž‰…(™*›—Ž¼F –—‡‰‹ ˆ›T‡ †D‹l∈…‰ŠŒ¥qR† ¹ ¹ ›T‡ ¬001 = 110 6∈ R ¹
· ‹Sº»†\Ž‰…(™*›—Ž£>œ*›T±¼F¹ –—‡‰‹ 001›T‡ ∧ 010 = 000 6∈ R ¹
· ‹Sº»†\Ž‰…(™*›—Ž{¢eŠ ½‰–—‹ …‰ŠŒ¥q† ¹ ›T‡ 001 ∨ 010 = 011 6∈ R ¹
·
= 000 6∈ R ¹
–—œŽ Ÿ ½‰› †D‹l010,
…†\Ž‰…100)
¨
·ƒ-–—†\‹ Ž‰…›¾‹††\Ž‰…(‘† Ž}šB¨eŽ‰… µ ¹ ›¶ˆ ‡}† †DYŽÀmaj(001,
x+y+z =1
›T…‰Š‘–—‹
‹· Sº» †\Ž‰¹ …(Pol(R
™*›—Žs¸ Ÿ )¥`›T=ŒŠ‘£eL† ¹ ›T‡ 000 6∈ R R¹ = [(x ∨ y) ∧ z]
· ‹S†\Ž‰º»…†\Ž‰§…(Ÿ ¥—™*›T›—ŒŽŠ‘£e¼F† –—¹ ‡‰ ‹ ›T‡ 111
· †\Ž‰…{£>œ*›T±¼F–—‡‰‹ ¹ ›T›T‡ ‡ 011∈ R∧ ¹101 = 001†D… 6∈ R ¹
· †\Ž‰…(¢eŠ ½cœe‹ …‰ŠŒ¥q† ¹ ›T‡ 111 ∨ m = 111 011 ∨ 101 = 111
∈R ¹
· ‹Sº»†\Ž‰…(™*›—Žs ›¾‹¹ † ›T‡ maj(011, 101, 111) = 111 ∈ R¨ †D… maj(m,
m, m ) = m ¹
·ƒ-–—‹
011 + 101 + 111 = 001 6∈ R
¹ Pol(R ) = S ¨ ¿ ›T‡ –—‹Ž}„\Á˜œ†D‹l… ¹ Pol(R , R ) = I †D…s’!“” (S) †\Ž‰…(™!–—Œšl‹–—ˆ$ŠY›T ¨
4
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0
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01
©
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