16. สรุปคณิตศาสตร์ ม.4 เทอม 1
16
1∈P(Z) .. ¼i´ (1 äÁ‹ãª‹e«μ¨ึ§o‹Ùã¹ P(Z) äÁ‹ä´Œ)
{1} ∈P(Z) .. ¶¡Ù e¾ÃÒa {1} ⊂ Z
∅ ∈P(Z) .. ¶Ù¡ e¾ÃÒa ∅ ⊂ Z
{∅} ∈P(Z) .. ¶¡Ù e¾ÃÒa {∅} ⊂ Z
{2, 3} ∈P(Z) .. ¼i´ e¾ÃÒa {2, 3} ⊄ Z
{{2,3}} ∈P(Z) .. ¶¡Ù e¾ÃÒa {{2,3}} ⊂ Z
18. ¡ÒÃæÊ´§e«μ´ŒÇÂæ¼¹ÀÒ¾¢o§eǹ¹æÅaooÂeÅoà ¨aãËŒ
eo¡À¾ÊaÁ¾a·¸e»š¹¡ÃoºÊÕèeËÅÕèÂÁ·ÕèãË
‹·ÕèÊu´ ÀÒÂ㹺Ãèu
ÃÙ»»´ (ǧ¡ÅÁ ǧÃÕ ÊÒÁeËÅÕèÂÁ ÏÅÏ) ·Õè㪌淹¢oºe¢μ
¢o§e«μ A, B, C μ‹Ò§æ
e«μ 2 e«μ (eª‹¹ A, B) ¨ae¡ÕèÂÇ¢Œo§¡a¹ä´Œ 5 Ãٻ溺´a§¹Õé
A B A B
äÁ‹ÁÕÊÁÒªi¡Ã‹ÇÁ¡a¹eÅ ÁÕÊÁÒªi¡Ã‹ÇÁ¡a¹ºÒ§Ê‹Ç¹
A
B
B
A
A e»š¹Êaºe«μ¢o§ B B e»š¹Êaºe«μ¢o§ A
17. พื้นฐาน บทที่ 1 เซต
17
A B
A æÅa B e·‹Ò¡a¹
** o´Â·aèÇ件ŒÒäÁ‹·ÃÒºÃٻ溺ªa´e¨¹ ¤ÇÃe¢Õ¹漹ÀÒ¾
ãËŒÁÕÊÁÒªi¡Ã‹ÇÁ¡a¹ºÒ§Ê‹Ç¹¡‹o¹ (e»š¹ÃÙ»ÁÒμðҹ) æÅÇŒ
¨ึ§oÒÈa¢ŒoÁÙÅoืè¹æ e¾ืèoãËŒ·ÃÒº¨íҹǹÊÁÒªi¡¢o§æμ‹Åa
ªié¹Ê‹Ç¹ «ึè§oÒ¨¾ºÇ‹ÒºÒ§ªié¹Ê‹Ç¹äÁ‹ÁÕÊÁÒªi¡eÅ¡çe»š¹ä»ä´ Œ
19. ¡ÒôíÒe¹i¹¡ÒÃe¡ÕèÂÇ¡aºe«μ 䴌桋 ÂÙe¹Õ¹, oi¹eμoÃe«¡
ªa¹, ¤oÁ¾ÅÕeÁ¹μ, æÅa¼Åμ‹Ò§¢o§e«μ e»š¹¡Ò÷íÒãËŒe¡i´
e«μãËÁ‹¢ึ鹨ҡe«μ·ÕèÁÕoÂÙ‹e´iÁ
ÂÙe¹Õ¹ ... ÂÙe¹Õ¹¢o§ A æÅa B e¢Õ¹e»š¹ A ∪ B
¤ืoe«μ¢o§ÊÁÒªi¡·aé§ËÁ´¢o§ A ¡aº B
(e·Õºe»š¹¤íÒÀÒÉÒä·Âä´ŒÇ‹Ò “A ËÃืo B”)
oi¹eμoÃe«¡ªa¹ ... e¢Õ¹e»š¹ A ∩ B ËÃืo AB
¤ืoe«μ¢o§ÊÁÒªi¡μaÇ·Õè»ÃÒ¡¯«éíÒ¡a¹ã¹ A æÅa B
(e·Õºe»š¹¤íÒÀÒÉÒä·Âä´ŒÇ‹Ò “A æÅa B”)
μÇaoÂÒ‹§ ¶ÒŒ A = {2, 3,5, 7}
æÅa B = {∅,1,2, {3, 4},5}
18. สรุปคณิตศาสตร์ ม.4 เทอม 1
18
¨aä´Œ A ∪ B = {∅,1, 2, 3,5, 7,{3, 4}}
æÅa A ∩ B = {2, 5}
¢oŒÊa§e¡μ A ∪ B = B ∪ A , A ∩ B = B ∩ A eÊÁo
æÅa¤‹Ò¢o§ n(A ∪B) + n(A ∩ B) = n(A) + n(B)
eÊÁo
¤oÁ¾ÅÕeÁ¹μ ... ¤oÁ¾ÅÕeÁ¹μ¢o§ A e¢Õ¹e»š¹ A'
¤ืoe«μ¢o§ÊÁÒªi¡·ÕèeËÅืoã¹ U «ึè§äÁ‹ä´ŒoÂÙ‹ã¹ A
¼Åμ‹Ò§ ... e«μ B − A ¤ืoe«μ¢o§ÊÁÒªi¡·Õèo‹Ùã¹ B æμ‹äÁ‹
o‹Ùã¹ A ... ÊÒÁÒöe¢Õ¹䴌oÕ¡æººÇ‹Ò B ∩ A'
μÇaoÂÒ‹§ ¶ÒŒ A = {2, 3,5, 7}
æÅa B = {∅,1,2, {3, 4},5}
o´Â·Õè U = {∅,1, 2,3, 4,5,6, 7,{1, 2}, {3, 4},(5,6)}
¨aä´Œ A ' = {∅,1, 4,6, {1,2}, {3, 4},(5,6)}
æÅa B ' = {3, 4,6,7, {1, 2},(5,6)}
¢oŒÊa§e¡μ ¤‹Ò¢o§ n(A) + n(A') = n(U) eÊÁo
μÇaoÂÒ‹§ ¶ÒŒ A = {2, 3,5, 7}
æÅa B = {∅,1,2, {3, 4},5}
¨aä´Œ B − A = {∅,1,{3, 4}}
æÅa A − B = {3,7}
19. พื้นฐาน บทที่ 1 เซต
19
20. ¤Ò‹¢o§ n(B − A) = n(B) − n(A ∩B) eÊÁo
æÅa¤‹Ò¢o§ n(A − B) = n(A) − n(A ∩B) eÊÁo
(ËŒÒÁ¤i´¨Ò¡ n(B) − n(A) ËÃืo n(A) − n(B) e¾ÃÒa
o´Â·aèÇä»Áa¡¨a¼i´)
μÇaoÂÒ‹§ ¶ÒŒ n(B) = 9 æÅa n(A) = 4
¨aÂa§¡Å‹ÒÇäÁ‹ä´ŒÇ‹Ò n(B − A) = 9 − 4 = 5
e¾ÃÒaÊÁÒªi¡¢o§ A ·aé§ÊÕèμaǹaé¹oÒ¨äÁ‹ä´ŒoÂÙ‹ã¹ B ·aé§ËÁ´
¨aμŒo§·ÃÒº¡‹o¹Ç‹Ò n(A ∩B) e·‹Ò¡aºe·‹Òã´
eª‹¹¶ŒÒ n(A ∩B) = 3 ¡ç¨aÊÃu»ä´ŒÇ‹Ò
n(B − A) = 9 − 3 = 6 æÅa n(A − B) = 4 − 3 = 1
μÇaoÂÒ‹§ ¶ÒŒ C = { ∅,0, 3,{0, 3}, {0, {3}} }
ãËŒËÒ¨íҹǹÊÁÒªi¡¢o§e«μ P(C) − C
äÁ‹¤Çäi´o´Â 25 − 5 = 27 e¾ÃÒaÊÁÒªi¡¢o§ C o‹Ùã¹
P(C) äÁ‹¶ึ§ 5 μaÇ æμ‹oÂÙ‹e¾Õ§ 2 μaÇ ¤ืo ∅ ¡aº {0,3}
´a§¹a鹤íÒμoº¨ึ§e·‹Ò¡aº 25 − 2 = 30
21. ¢ŒoÊa§e¡μ·Õè¤Ç÷ÃÒº
∅' = U A ∩A' = ∅
U ' = ∅ A ∪A' = U
A ⊂ (A ∪B) A − U = ∅
20. สรุปคณิตศาสตร์ ม.4 เทอม 1
20
ง
B ⊂ (A ∪B) U − A = A'
(A ∩B) ⊂ A A −∅ = A
(A ∩B) ⊂ B ∅ − A = ∅
ก ข ค
A B
U
A − A = ∅
** ¶ŒÒ A − B = ∅ æÅŒÇ äÁ‹¨íÒe»š¹·ÕèÇ‹Ò A = B
(¶ŒÒ A = B ‹oÁ·íÒãËŒ A − B = ∅ 湋¹o¹
æμ‹Âa§ÁաóÕoืè¹æ oÕ¡ ¤ืoeÁืèoäáçμÒÁ·Õè A ⊂ B ¡çä´Œ)
22. ÊÁºaμi¢o§¤oÁ¾ÅÕeÁ¹μ æÅa¡ÒÃ模樧
(A ∪B) ' = A' ∩B '
(A ∩B) ' = A' ∪B '
A ∩(B ∪ C) = (A ∩ B) ∪(A ∩ C)
A ∪(B ∩ C) = (A ∪ B) ∩(A ∪ C)
23. o¨·Â»˜ËÒ
·Õèe»š¹eËμu¡Òó ¨a㪌漹ÀÒ¾eǹ¹-ooÂ
eÅoà ª‹ÇÂ㹡Òäíҹdzªié¹Ê‹Ç¹μ‹Ò§æ æμ‹ã¹¢ŒoÊoºÁa¡
μaé§ã¨ãˌ㪌ÊÙμÃ㹡ÒÃËÒ¨íҹǹÊÁÒªi¡æμ‹Åaªié¹Ê‹Ç¹´a§¹Õé
** ÊíÒËÃaº 2 e«μ
n(A ∪B) = n(A) + n(B) − n(A ∩B)
41. พื้นฐาน บทที่ 3 จำนวนจริง
41
μÇaoÂÒ‹§ e«μμ‹o仹ÕéÁÕÊÁºaμi»´¡ÒúǡËÃืoäÁ‹ æÅaÁÕ
ÊÁºaμi»´¡ÒäٳËÃืoäÁ?‹
{−1, 0,1}
äÁ‹ÁÕÊÁºaμi»´¡Òúǡ (eª‹¹ 1 + 1 = 2 «ึè§äÁ‹o‹Ùã¹e«μ¹Õé)
æμ‹ÁÕÊÁºaμi»´¡Òäٳ (e¾ÃÒaäÁ‹Ç‹Ò¨a¹íÒ¨íҹǹã´ã¹e«μ¹Õé
ÁÒ¤Ù³¡a¹ ¼ÅÅa¾¸·Õèä´ŒÂa§¤§oÂÙ‹ã¹e«μ¹ÕéeÊÁo)
{0,−1,−2,−3,−4,...}
ÁÕÊÁºaμi»´¡Òúǡ (e¾ÃÒa¨íҹǹ 0 æÅa¨íҹǹeμçÁź
ºÇ¡¡a¹o‹ҧäáç‹oÁä´Œ¨íҹǹ 0 æÅa¨íҹǹeμçÁźeÊÁo)
æμ‹äÁ‹ÁÕÊÁºaμi»´¡Òäٳ (eª‹¹ (−1) ⋅ (−2) = 2 «ึè§äÁ‹o‹Ù
ã¹e«μ¹Õé)
e«μ¢o§¨íҹǹoμÃáÂa
äÁ‹ÁÕÊÁºaμi»´¡Òúǡ (eª‹¹ 2 + (− 2) = 0 )
æÅaäÁ‹ÁÕÊÁºaμi»´¡Òäٳ (eª‹¹ 2 ⋅ 2 = 2 )
7. “eo¡Åa¡É³” ¤ืo¨íҹǹ·Õèä»´íÒe¹i¹¡Òáaº¨íҹǹ a
ã´¡çμÒÁ æÅŒÇä´Œ¼ÅÅa¾¸ a e·‹Òe´iÁ
ËÃืo a ∗ e = e ∗ a = a (eÁืèo e ¤ืoeo¡Åa¡É³)
e¹ืèo§¨Ò¡ a + 0 = 0 + a = a
´a§¹aé¹eo¡Åa¡É³¡Òúǡ¢o§¨íҹǹ¨Ãi§ã´æ ¤ืo 0
42. สรุปคณิตศาสตร์ ม.4 เทอม 1
42
æÅae¹ืèo§¨Ò¡ a ⋅1 = 1⋅ a = a
´a§¹aé¹eo¡Åa¡É³¡Òäٳ¢o§¨íҹǹ¨Ãi§ã´æ ¤ืo 1
μÇaoÂÒ‹§ ¶ÒŒ¹iÂÒÁãËŒ x ∗ y = x + y − 2
ãËŒËÒeo¡Åa¡É³¢o§¡ÒôÒíe¹i¹¡Òù Õé
¨Ò¡ a ∗ e = a ¨aä´Œ a + e − 2 = a
¹a蹤ืo e = 2
æÅa¨Ò¡ e ∗ a = a ¨aä´Œ e + a − 2 = a
¹a蹤ืo e = 2 eª‹¹¡a¹
´a§¹aé¹ÊÃu»Ç‹Ò eo¡Åa¡É³¢o§¡ÒôÒíe¹i¹¡Òù¤Õéืo 2
μÇaoÂÒ‹§ ¶ÒŒ¹iÂÒÁãËŒ x ∗ y = x − y + 2
ãËŒËÒeo¡Åa¡É³¢o§¡ÒôÒíe¹i¹¡Òù Õé
¨Ò¡ a ∗ e = a ¨aä´Œ a − e + 2 = a
¹a蹤ืo e = 2
æÅa¨Ò¡ e ∗ a = a ¨aä´Œ e − a + 2 = a
¹a蹤ืo e = 2a − 2
¾ºÇ‹Òeo¡Åa¡É³·ÕèËÒä´Œ¨Ò¡Êo§Çi¸ÕÁÕ¤‹ÒäÁe‹·‹Ò¡a¹ ´a§¹a鹡ÒÃ
´íÒe¹i¹¡ÒÃã¹¢Œo¹Õé “äÁ‹ÁÕeo¡Åa¡É³”
** ¡ÒôíÒe¹i¹¡ÒÃã´¨aÁÕeo¡Åa¡É³ä´Œ¹aé¹ ¨aμŒo§ÁÕÊÁºaμi
¡ÒÃÊÅaº·Õè¡‹o¹ e¾ÃÒa a ∗ e μŒo§e·‹Ò¡aº e ∗ a ´ŒÇÂ
43. พื้นฐาน บทที่ 3 จำนวนจริง
43
8. “oi¹eÇoÃÊ (μaǼ¡¼a¹) ¢o§ a” ¤ืo¨íҹǹ·Õèä»
´íÒe¹i¹¡Òáaº¨íҹǹ a æÅŒÇä´Œ¼ÅÅa¾¸e»š¹eo¡Åa¡É³
ËÃืo a ∗ i = i ∗ a = e (eÁืèo i ¤ืooi¹eÇoÃÊ)
e¹ืèo§¨Ò¡ a + (−a) = (−a)+ a = 0
´a§¹aé¹oi¹eÇoÃÊ¡Òúǡ¢o§¨íҹǹ¨Ãi§ a ¤ืo –a
æÅae¹ืèo§¨Ò¡ a ⋅(1/a) = (1/a) ⋅ a = 1
´a§¹aé¹oi¹eÇoÃÊ¡Òäٳ¢o§¨íҹǹ¨Ãi§ a ¤ืo 1/a
(¡eÇŒ¹eÁืèo a = 0 ¨aäÁ‹ÁÕoi¹eÇoÃÊ¡Òä³Ù)
** ¤‹Ò 1/a ÊÒÁÒöe¢Õ¹e»š¹ a −1 ä´Œ´ŒÇÂ
(o‹Ò¹Ç‹Ò “a ¡¡íÒÅa§ÅºË¹ึ觔 ËÃืo “a oi¹eÇoÃÊ” ¡çä´Œ)
μÇaoÂÒ‹§ ¶ÒŒ¹iÂÒÁãËŒ x ∗ y = x + y − 2
ãËŒËÒoi¹eÇoÃÊ¢o§ a ÊíÒËÃaº¡ÒôíÒe¹i¹¡Òù Õé
¨Ò¡¢Œo·ÕèæÅŒÇ eo¡Åa¡É³¢o§¡ÒôÒíe¹i¹¡Òù¤Õéืo 2
´a§¹aé¹ a ∗ i = 2 ¨aä´Œ a + i − 2 = 2
¹a蹤ืo i = 4 − a
(ËÃืo¤i´¨Ò¡ i ∗ a = 2 ¡ç¨aä´Œ i = 4 − a eª‹¹¡a¹)
ÊÃu»Ç‹Òoi¹eÇoÃÊ¢o§ a ã¹¢Œo¹Õé¤ืo 4 − a
μÇaoÂÒ‹§ ¶ÒŒ¹iÂÒÁãËŒ x ∗ y = x − y + 2
¡ÒôíÒe¹i¹¡ÒùÕé¨aäÁ‹ÁÕoi¹eÇoÃÊ e¾ÃÒaäÁ‹ÁÕeo¡Åa¡É³
44. สรุปคณิตศาสตร์ ม.4 เทอม 1
44
9. o´ÂÊÃu» Ãaºº¨íҹǹ¨Ãi§ÁÕÊÁºaμi 11 o‹ҧ ´a§¹Õé
(1) ÊÁºaμi»´¢o§¡Òúǡ
¶ŒÒ a æÅa b e»š¹¨íҹǹ¨Ãi§æÅŒÇ a+b e»š¹¨íҹǹ¨Ãi§
(2) ÊÁºaμi»´¢o§¡Òäٳ
¶ŒÒ a æÅa b e»š¹¨íҹǹ¨Ãi§æÅŒÇ ab e»š¹¨íҹǹ¨Ãi§
(3) ÊÁºaμi¡ÒÃÊÅaº·Õè¢o§¡Òúǡ
a + b = b + a
(4) ÊÁºaμi¡ÒÃÊÅaº·Õè¢o§¡Òäٳ
a b = b a
(5) ÊÁºaμi¡ÒÃe»ÅÕ蹡Åu‹Á¢o§¡Òúǡ
a + (b + c) = (a + b) + c
(6) ÊÁºaμi¡ÒÃe»ÅÕ蹡Åu‹Á¢o§¡Òäٳ
a (b c) = (a b) c = a b c
(7) ÊÁºaμi¡ÒÃ模樧
a (b + c) = a b + a c æÅa (a + b) c = a c + b c
(8) ÊÁºaμi¡ÒÃÁÕeo¡Åa¡É³¡Òúǡ
e¹ืèo§¨Ò¡ a + 0 = 0 + a = a
´a§¹aé¹eo¡Åa¡É³¡Òúǡ¢o§¨íҹǹ¨Ãi§ã´æ ¤ืo 0
(9) ÊÁºaμi¡ÒÃÁÕeo¡Åa¡É³¡Òäٳ
e¹ืèo§¨Ò¡ a ⋅1 = 1⋅ a = a
´a§¹aé¹eo¡Åa¡É³¡Òäٳ¢o§¨íҹǹ¨Ãi§ã´æ ¤ืo 1
(10) ÊÁºaμi¡ÒÃÁÕoi¹eÇoÃÊ¡Òúǡ
e¹ืèo§¨Ò¡ a + (−a) = (−a)+ a = 0
´a§¹aé¹oi¹eÇoÃÊ¡Òúǡ¢o§¨íҹǹ¨Ãi§ a ¤ืo –a
45. พื้นฐาน บทที่ 3 จำนวนจริง
45
(11) ÊÁºaμi¡ÒÃÁÕoi¹eÇoÃÊ¡Òäٳ
e¹ืèo§¨Ò¡ a ⋅(1/a) = (1/a) ⋅ a = 1
´a§¹aé¹oi¹eÇoÃÊ¡Òäٳ¢o§¨íҹǹ¨Ãi§ a ¤ืo 1/a
(¡eÇŒ¹eÁืèo a = 0 ¨aäÁ‹ÁÕoi¹eÇoÃÊ¡Òä³Ù)
10. ¡Òäíҹdze¡ÕèÂÇ¡aºeÈÉʋǹ
(1) ¡ÒúǡeÈÉʋǹ a + d = ac + bd
b c bc
(2) ¡ÒäٳeÈÉʋǹ a ⋅ d = ad
b c bc
(3) eÈÉʋǹ«Œo¹ a b = a
c bc æÅa a = ac
b c b
(4) ¡ÒÃËÒÃeÈÉʋǹ a b = ad
c d bc
(5) oi¹eÇoÃÊ¡Òäٳ¢o§eÈÉʋǹ
− ⎛ ⎞ = ⎜ ⎟
⎝ ⎠
1 a b
b a
11. ÊÁ¡Òà ¤ืo»Ãao¤·ÕèÁÕμaÇæ»ÃæÅa¡Å‹ÒǶึ§¡ÒÃe·‹Ò¡a¹
• ¡Òà “æ¡ŒÊÁ¡ÒÔ ¤ืo¡ÒÃËÒ¤‹Ò¢o§μaÇæ»Ã·Õè·íÒãËŒ
»Ãao¤¹aé¹e»š¹¨Ãi§.. oÒ¨¡Å‹ÒÇÇ‹Òe»š¹¡ÒÃËÒ “e«μ¤íÒμoº
¢o§ÊÁ¡ÒÔ ËÃืo¡ÒÃËÒ “ÃÒ¡¢o§ÊÁ¡ÒÔ ¡çä´Œ
46. สรุปคณิตศาสตร์ ม.4 เทอม 1
46
** ¤íÒÇÒ‹ “ÃÒ¡¢o§ÊÁ¡ÒÔ æ»ÅÇ‹Ò¤íÒμoº¢o§ÊÁ¡ÒÃ
(äÁ‹ä´ŒËÁÒ¤ÇÒÁe¡ÕèÂÇ¡aº¡Òöo´ÃÙŒ·)
μÇaoÂÒ‹§ ãËŒËÒe«μ¤Òíμoº æÅa¼ÅºÇ¡¢o§ÃÒ¡¢o§ÊÁ¡ÒÃ
x2 − 3x = 0
e¹ืèo§¨Ò¡ x2 − 3x = (x)(x − 3) = 0
ËÁÒ¤ÇÒÁÇÒ‹ x = 0 ËÃืo x − 3 = 0
æÊ´§Ç‹Ò ÃÒ¡¢o§ÊÁ¡ÒÃ䴌桋 0 ¡aº 3
´a§¹aé¹e«μ¤íÒμoº¤ืo {0,3}
æÅa¼ÅºÇ¡ÃÒ¡¢o§ÊÁ¡Òà e·‹Ò¡aº 0 + 3 = 3
12. ¢Œo¤ÇÃÃaÇa§ã¹¡ÒÃæ¡ŒÊÁ¡ÒÃã´æ
• ¡ÒúǡËÃืoź·aé§Êo§¢ŒÒ§ (ŒҢŒÒ§ºÇ¡Åº) æÅa¡ÒÃ
μa´oo¡ÊíÒËÃaº¡ÒúǡËÃืoź ·íÒä´ŒeÊÁo
a = b → a ± c = b ± c eÊÁo
a ± c = b ± c → a = b eÊÁo
• ¡Òäٳ·aé§Êo§¢ŒÒ§ (ŒҢŒÒ§¤Ù³) ·íÒä´ŒeÊÁo ¡ÒÃ
ËÒ÷aé§Êo§¢ŒÒ§ (ŒҢŒÒ§ä»ËÒÃ) μaÇËÒÃËŒÒÁe»š¹ 0
a = b → a c = b c eÊÁo
a = b → a /c = b /c eÁืèo c ≠ 0
• ¡ÒÃμa´oo¡ÊíÒËÃaº¡Òäٳ ·íÒä´ŒeÁืèoÁaè¹ã¨Ç‹ÒeÅ¢·Õèμa´
oo¡·aé§Êo§¢ŒÒ§äÁ‹ãª‹ 0
a c = b c → a = b eÁืèo c ≠ 0
47. พื้นฐาน บทที่ 3 จำนวนจริง
47
• ¡Òá¡íÒÅa§Êo§·aé§Êo§¢ŒÒ§ ·íÒä´ŒeÊÁo
æμ‹¡ÒÃμa´¡íÒÅa§Êo§oo¡ ¨aÁռŠ2 ¡Ã³Õ ¤ืoÊo§¢ŒÒ§
e·‹Ò¡a¹ ËÃืoÊo§¢ŒÒ§e»š¹μi´Åº¢o§¡a¹æÅa¡a¹
a = b → a2 = b2 eÊÁo
a2 = b2 → a = b ËÃืo a = −b
μÇaoÂÒ‹§ ãËŒËÒe«μ¤Òíμoº¢o§ÊÁ¡Òà x2 = 3x
¶ŒÒμa´ x oo¡Ë¹ึè§μaÇ·aé§Êo§¢ŒÒ§ ¡ÅÒÂe»š¹ x = 3
¨aä´Œe«μ¤íÒμoº¤ืo {3} æμ‹e»š¹¤íÒμoº·Õè¼i´!
Çi¸Õ·Õè¶Ù¡ ¨aμŒo§ÂŒÒÂÁÒź¡a¹´a§¹Õé.. x2 − 3x = 0
¨aä´Œ (x)(x − 3) = 0
ËÁÒ¤ÇÒÁÇÒ‹ x = 0 ËÃืo x − 3 = 0
´a§¹aé¹e«μ¤íÒμoº·Õè¶Ù¡μŒo§¤ืo {0,3}
¨aeËç¹Ç‹Ò¶ŒÒμa´ x oo¡·aé§Êo§¢ŒÒ§ ¨aÅืÁ¤íÒμoº x = 0
** ÊÃu»¤ืo ËÒ¡o¨·Â¡íÒ˹´Ç‹Ò x äÁ‹e»š¹Èٹ ÊÒÁÒö
μa´oo¡ä´Œ æ싶ŒÒ x oÒ¨e»š¹ÈÙ¹Âä´Œ ËŒÒÁμa´oo¡!
13. ¾Ëu¹ÒÁ ¤ืoÃٻ溺ª¹i´Ë¹ึ觷ҧ¤³iμÈÒÊμÃ
¾Ëu¹ÒÁ·ÕèÁÕ x e»š¹μaÇæ»ÃμaÇe´ÕÂÇ ¨aoÂÙ‹ã¹ÃÙ»
−
− n + n 1+ + +
n n1 1 0 a x a x ... a x a
48. สรุปคณิตศาสตร์ ม.4 เทอม 1
48
(a ·§aéËÁ´e»¹š¤Ò¤‹§·Õè eÃÕ¡ÇÒ ‹“ÊaÁ»ÃaÊi·¸iì”
æÅa n ¤oื¨íҹǹ¹ºaã´æ)
¡Ò÷íÒ¾Ëu¹ÒÁãËoÂٌ㋹ÃÙ»¼Å¤Ù³¢o§¾Ëu¹ÒÁ·ÕèÁÕ´Õ¡ÃÕμèíÒŧ
eÃÕ¡ÇÒ ‹“¡ÒÃæ¡μaÇ»Ãa¡oº”
14. ÊÁºaμi·ÕèÊíÒ¤aã
¹¡ÒÃæ¡ÊÁ¡ŒÒáíÒÅa§Êo§¤o ืËÒ¡
a b = 0 æÅÇŒ¨aä´ÇŒÒ ‹a = 0 ËÃืo b = 0
ÊÁ¡ÒáíÒÅa§Êo§ã¹Åa¡É³a x2 + Bx + C = 0
¤ÇÃæ¡μaÇ»Ãa¡oºãËoÂŒÙ㋹ÃÙ» (x + D)(x + E) = 0
e¾èoื¨aä´·ŒÃÒºÇÒ ‹¤íÒμoº¢o§ÊÁ¡ÒÃä´æ¡Œ‹ x = −D ËÃืo
x = −E
μÇaoÂÒ‹§ ãËËÒeŒ«μ¤Òíμoº¢o§ÊÁ¡Òà x2 − 6x + 5 = 0
æ¡μaÇ»Ãa¡oºä´eŒ»š¹ (x − 5)(x −1) = 0
´§a¹¹ aéx − 5 = 0 ËÃืo x −1 = 0
¤íÒμoº¢o§ÊÁ¡ÒÃä´æ¡Œ‹ x = 5 ËÃืo x = 1
æÅae«μ¤íÒμoº¤o ื{5,1}
2 μÇaoÂÒ‹§ ãËËÒeŒ«μ¤Òíμoº¢o§ÊÁ¡Òà 4 − x
=
x
2
¹íÒ 2 ¤Ù³·aé§Êo§¢ŒÒ§¢o§ÊÁ¡Òà e¾ืèoäÁ‹ãËŒÁÕeÈÉʋǹ
49. พื้นฐาน บทที่ 3 จำนวนจริง
49
¨aä´Œe»š¹ 8 − x2 = 2x
¨Ò¡¹aé¹ÂŒÒ¢ŒÒ§ãËŒÁÕ½˜›§Ë¹ึè§e»š¹ 0
¨aä´Œ 0 = x2 + 2x − 8 ¹a蹤ืo 0 = (x + 4)(x − 2)
´a§¹aé¹ x + 4 = 0 ËÃืo x − 2 = 0
¤íÒμoº¢o§ÊÁ¡ÒÃ䴌桋 x = −4 ËÃืo x = 2
¨ึ§ä´ŒÇ‹Ò e«μ¤Òíμoº¤ืo {−4, 2}
μÇaoÂÒ‹§ ãËŒËÒe«μ¤Òíμoº¢o§ÊÁ¡Òà x2 − 3 = 0
æ¡μaÇ»Ãa¡oºä´Œe»š¹ (x − 3)(x + 3) = 0
´a§¹aé¹ x − 3 = 0 ËÃืo x + 3 = 0
¤íÒμoº¢o§ÊÁ¡ÒÃ䴌桋 x = 3 ËÃืo x = − 3
æÅae«μ¤íÒμoº¤ืo { 3,− 3}
(ËÃืooÒ¨e¢Õ¹e»š¹ {± 3 } ¡çä´Œ)
** ¶ŒÒe»ÅÕè¹o¨·Âe»š¹ x2 + 3 = 0 ¨aäÁ‹ÁÕ¤íÒμoº·Õèe»š¹
¨íҹǹ¨Ãi§ e¹ืèo§¨Ò¡¨aæ¡μaÇ»Ãa¡oºäÁ‹ä´Œ
(ËҡŒҢŒÒ§¨a¾ºÇ‹Òä´ŒÊÁ¡Òà x2 = −3 æÊ´§Ç‹Ò¤‹Ò x
¹Õée»š¹ÃÒ¡·ÕèÊo§¢o§ –3 ¨ึ§äÁ‹ãª‹¨íҹǹ¨Ãi§)
15. ÊÁ¡ÒáíÒÅa§Êo§ ÁÕÃÙ»·aèÇä»e»š¹ Ax2 +Bx + C = 0
eÁืèoæ¡μaÇ»Ãa¡oºe»š¹ (Dx + E)(Fx + G) = 0 ¨a·ÃÒº
Ç‹Ò¤íÒμoº¢o§ÊÁ¡ÒÃ䴌桋 = − E
x
ËÃืo x
= − G
D F
59. พื้นฐาน บทที่ 3 จำนวนจริง
59
μÇaoÂÒ‹§ oÊÁ¡Òà −1< x < 2 e¢Õ¹eÊŒ¹¨íҹǹ䴌´a§¹Õé
-1 2
æÅae¢Õ¹e»š¹ª‹Ç§ä´Œe»š¹ (−1,2]
o‹Ò¹Ç‹Ò “ª‹Ç§e»´ -1 ¶ึ§»´ 2”
21. e¹ืèo§¨Ò¡ “ª‹Ç§” ¤ืoe«μª¹i´Ë¹ึè§ (e«μ·ÕèÁÕÊÁÒªi¡e»š¹
¨íҹǹ¨Ãi§æÅaÁÕ¤‹Òμ‹oe¹ืèo§) ´a§¹aé¹ÊÒÁÒö¹íÒª‹Ç§Êo§ª‹Ç§
ÁÒÂÙe¹Õ¹ oi¹eμoÃe«¤ ËÃืoź¡a¹¡çä´Œ æÅaËÒ¤oÁ¾ÅÕ
eÁ¹μ¢o§ª‹Ç§¡çä´Œ o´Â¹iÂÁ¾i¨ÒóҨҡeÊŒ¹¨íҹǹ
** ª‹Ç§·ÕèãË
‹·ÕèÊu´¤ืoe«μ = −∞ ∞ R ( , )
æÅaª‹Ç§·ÕèeÅç¡·ÕèÊu´¤ืo ∅
μÇaoÂÒ‹§ ¡Òí˹´ A = [1, 4] æÅa B = (−2,3)
ãËŒËÒ A ∩ B æÅa A ∪ B æÅa (A ∪B) '
¨aä´Œ A ∩ B = [1,3) ´a§ÃÙ»
-2 1 3 4
60. สรุปคณิตศาสตร์ ม.4 เทอม 1
60
æÅaä´Œ A ∪ B = (−2, 4] ´a§ÃÙ»
-2 1 3 4
´a§¹aé¹ (A ∪B) ' = (−∞,−2]∪(4,∞)
-2 1 3 4
μÇaoÂÒ‹§ ¡Òí˹´ A = [−2,∞) æÅa B = (−2,3]
ãËŒËÒ A − B æÅa B − A
¨aä´Œ A − B = {2} ∪(3,∞) ´a§ÃÙ»
-2 1 3
æÅaä´Œ B − A = ∅
22. ¢Œo¤ÇÃÃaÇa§ã¹¡ÒÃæ¡ŒoÊÁ¡ÒÃã´æ
• ¡ÒúǡËÃืoź·aé§Êo§¢ŒÒ§ (ŒҢŒÒ§ºÇ¡Åº) æÅa¡ÒÃ
μa´oo¡ÊíÒËÃaº¡ÒúǡËÃืoź ·íÒä´ŒeÊÁo
a > b → a ± c > b ± c eÊÁo
a ± c > b ± c → a > b eÊÁo
61. พื้นฐาน บทที่ 3 จำนวนจริง
61
• ¡ÒäٳËÃืoËÒ÷aé§Êo§¢ŒÒ§ (ŒҢŒÒ§¤Ù³ËÒÃ)
¨aμŒo§ÃaÇa§eÃืèo§¡ÒÃe»ÅÕè¹e¤Ãืèo§ËÁÒÂ
(¶ŒÒeÅ¢·ÕèÂŒÒÂe»š¹¤‹Òμi´Åº μŒo§¾Åi¡´ŒÒ¹e¤Ãืèo§ËÁÒÂ)
a > b → a c > b c eÁืèo c > 0
a > b → a c < b c eÁืèo c < 0
a c > b c → a > b eÁืèo c > 0
a c > b c → a < b eÁืèo c < 0
• ¡Òá¡íÒÅa§Êo§·aé§Êo§¢ŒÒ§ ·íÒä´ŒeÁืèoÁaè¹ã¨Ç‹Òe»š¹ºÇ¡
·aé§Êo§¢ŒÒ§ ËÃืoμi´Åº·aé§Êo§¢ŒÒ§e·‹Ò¹aé¹
(o´Â¡Ã³Õμi´ÅºμŒo§¾Åi¡´ŒÒ¹e¤Ãืèo§ËÁÒ´ŒÇÂ)
a > b → a2 > b2 eÁืèo a,b > 0
a > b → a2 < b2 eÁืèo a,b < 0
μÇaoÂÒ‹§ ¨Ò¡oÊÁ¡Òà −8 < 1 − 3x <13
¹íÒ 1 źoo¡ ä´Œe»š¹ −9 < −3x <12
¨Ò¡¹aé¹ËÒôŒÇ -3 ¨aä´Œ 3 > x > −4
** μŒo§¾Åi¡e¤Ãืèo§ËÁÒÂe¾ÃÒa¤‹Ò·Õè¹íÒÁÒËÒÃe»š¹¤‹Òμi´Åº
´a§¹aé¹e«μ¤íÒμoº¤ืo { x | − 4 < x < 3 }
ËÃืoe¢Õ¹e»š¹ª‹Ç§ [−4,3)
23. ¡ÒÃæ¡Œ (ËÃืoËÒ¤íÒμoº¢o§) oÊÁ¡ÒáíÒÅa§Êo§ e¾ืèo
¤ÇÒÁÊa´Ç¡¤ÇÃ㪌e·¤¹i¤´a§¹ Õé
81. เพิ่มเติม บทที่ 1 ตรรกศาสตร์เบื้องต้น
81
3. μÒÃÒ§μ‹o仹Õé æÊ´§¼Å·Õèä´Œ¨Ò¡¡ÒÃeªืèoÁ»Ãa¾¨¹´ŒÇÂ
“æÅa”, “ËÃืo”, “¶ŒÒ..æŌǔ, “¡çμ‹oeÁืèo”
p q p æÅa q
( p ∧ q )
p ËÃืo q
( p ∨ q )
¶ŒÒ p æÅŒÇ q
( p→q )
p ¡çμ‹oeÁืèo q
( p↔q )
äÁ‹ p
( ~p )
T T T T T T F
T F F T F F F
F T F T T F T
F F F F T T T
¡ÒÃeªืèoÁ´ŒÇ “æÅa” ÁաóÕe´ÕÂÇ·Õèe»š¹¨Ãi§ ¤ืo T ∧ T
¡ÒÃeªืèoÁ´ŒÇ “ËÃืo” ÁաóÕe´ÕÂÇ·Õèe»š¹e·ç¨ ¤ืo F ∨ F
¡ÒÃeªืèoÁ´ŒÇ “¶ŒÒ..æŌǔ ÁաóÕe´ÕÂÇ·Õèe»š¹e·ç¨¤ืo T→F
ʋǹ¡ÒÃeªืèoÁ´ŒÇ “¡çμ‹oeÁืèo” ¶ŒÒ¤‹Ò¤ÇÒÁ¨Ãi§eËÁืo¹¡a¹¨a
ãËŒ¼Åe»š¹¨Ãi§ μ‹Ò§¡a¹¨aãËŒ¼Åe»š¹e·ç¨
** μaÇeªืèoÁeËÅ‹Ò¹ÕéÁÕÊÁºaμi¡ÒÃÊÅaº·Õè ¡eÇŒ¹ “¶ŒÒ..æŌǔ «ึè§
äÁ‹ÊÒÁÒöÊÅaº·Õèä´Œ
4. μÒÃÒ§·ÕèæÊ´§Ãٻ溺¢o§¤‹Ò¤ÇÒÁ¨Ãi§·Õèe»š¹ä»ä´¤ŒÃº
·u¡¡Ã³Õ (eª‹¹ã¹¢Œo·ÕèæÅŒÇ) eÃÕÂ¡Ç‹Ò “μÒÃÒ§¤‹Ò¤ÇÒÁ¨Ãi§”
• ¶ŒÒÁÕ 1 »Ãa¾¨¹¨ae»š¹ä»ä´Œ 2 ¡Ã³Õ, ¶ŒÒÁÕ 2
»Ãa¾¨¹ ¨ae»š¹ä»ä´Œ 4 ¡Ã³Õ, ..ËÃืo¶ŒÒÁÕ n »Ãa¾¨¹
¨ae»š¹ä»ä´Œ 2n ¡Ã³Õ¹aè¹eo§
82. สรุปคณิตศาสตร์ ม.4 เทอม 1
82
μÇaoÂÒ‹§ ¡Òí˹´ãËŒ»Ãa¾¨¹ p, r ÁÕ¤‹Ò¤ÇÒÁ¨Ãi§e»š¹¨Ãi§
æÅa»Ãa¾¨¹ q ÁÕ¤‹Ò¤ÇÒÁe»š¹¨Ãi§e»š¹e·ç¨ ãËŒËÒ¤‹Ò¤ÇÒÁ
¨Ãi§¢o§Ãٻ溺»Ãa¾¨¹μ‹o仹Õé
• [(q→p) ∧r]↔r
¨aä´Œ¤‹Ò¤ÇÒÁ¨Ãi§e»š¹ [(F→T) ∧ T]↔ T
¹a蹤ืo [T ∧ T]↔ T ¹a蹤ืo T ↔ T ¡ç¤ืo T
´a§¹aé¹ Ãٻ溺»Ãa¾¨¹¹ÕéÁÕ¤‹Ò¤ÇÒÁ¨Ãi§e»š¹ “¨Ãi§”
• [(p ∧ q)→~r]→[(~p ∨ q)↔r]
¨aä´Œ¤‹Ò¤ÇÒÁ¨Ãi§e»š¹
[(T ∧ F)→~T]→[(~ T ∨F)↔T]
¹a蹤ืo [F→F]→[F↔T] ¹a蹤ืo T →F ¡ç¤ืo F
´a§¹aé¹ Ãٻ溺»Ãa¾¨¹¹ÕéÁÕ¤‹Ò¤ÇÒÁ¨Ãi§e»š¹ “e·ç¨”
5. eÁืèo¤uŒ¹e¤Â¡aºÅa¡É³a¢o§μaÇeªืèoÁ·aé§ÊÕèæÅŒÇ ¨a·íÒãËŒ
ÊÃu»¼Å¡Ã³Õ·aèÇä»ä´Œ´a§¹Õé (äÁÇ‹‹Ò p ¨ae»š¹»Ãa¾¨¹ã´æ)
• e¤Ãืèo§ËÁÒ “æÅa”
T ∧ p ≡ p , F ∧ p ≡ F , p ∧ p ≡ p , p ∧ ~p ≡ F
• e¤Ãืèo§ËÁÒ “ËÃืo”
T ∨ p ≡ T , F ∨ p ≡ p , p ∨ p ≡ p , p ∨ ~p ≡ T
85. เพิ่มเติม บทที่ 1 ตรรกศาสตร์เบื้องต้น
85
μÇaoÂÒ‹§ ¾i¨ÒóҤ‹Ò¤ÇÒÁ¨Ãi§æμ‹Åa¡Ã³Õ¢o§Ã»Ù溺
»Ãa¾¨¹ p →~q æÅa ~(p ∧ q) o´ÂÊÌҧμÒÃÒ§¤‹Ò
¤ÇÒÁ¨Ãi§´a§¹ Õé
p q p →~q ~(p ∧ q)
T T F F
T F T T
F T T T
F F T T
¾ºÇ‹Ò¤‹Ò¤ÇÒÁ¨Ãi§¢o§Ãٻ溺»Ãa¾¨¹ p →~q æÅa
~(p ∧ q) eËÁืo¹¡a¹eÊÁo·u¡æ ¡Ã³Õ.. æÊ´§Ç‹ÒûÙ溺·aé§
Êo§¹Õé “ÊÁÁÙÅ¡a¹”
7. Ãٻ溺»Ãa¾¨¹·ÕèÊÁÁÙÅ¡a¹ (溺¾ืé¹°Ò¹·Õè¤Ç÷ÃÒº)
• ¡ÒÃ模樧
p ∨(q ∧r) ≡ (p ∨ q)∧(p ∨ r)
p ∧(q∨ r) ≡ (p ∧ q)∨(p ∧r)
• ¡ÒÃe»ÅÕè¹μaÇeªืèoÁ
¶ŒÒ-æÅŒÇ.. p→q ≡ ~p ∨ q ≡ ~ q→~p
¡çμ‹oeÁืèo.. p↔q ≡ (p→q)∧(q→p)
88. สรุปคณิตศาสตร์ ม.4 เทอม 1
88
«ึè§ÁÕ¹ieʸe»š¹ ~p ∧ ~q
..¹a蹤ืo “ÊÁªÒ¹o¹äÁ‹ËÅaºæÅaäÁ‹ËiÇ”
8. ËÒ¡Ãٻ溺¢o§»Ãa¾¨¹ãËŒ¤‹Òe»š¹¨Ãi§eÊÁo (ÊÌҧ
μÒÃÒ§¤‹Ò¤ÇÒÁ¨Ãi§æÅÇŒ¾ºÇ‹Ò¼Åe»š¹¨Ãi§·u¡Ã³Õ) ¨aeÃÕ¡
Ãٻ溺¹aé¹Ç‹Òe»š¹ “Êa¨¹iÃa¹´Ã”
μÇaoÂÒ‹§ ¾i¨ÒóҤ‹Ò¤ÇÒÁ¨Ãi§æμ‹Åa¡Ã³Õ¢o§Ã»Ù溺
»Ãa¾¨¹ [(p→q) ∧(q→r)]→(p→r) o´ÂÊÌҧ
μÒÃÒ§¤‹Ò¤ÇÒÁ¨Ãi§´a§¹ Õé
p q r [(p→q) ∧(q→r)]→(p→r)
T T T T
T T F T
T F T T
T F F T
F T T T
F T F T
F F T T
F F F T
¾ºÇ‹Ò¤‹Ò¤ÇÒÁ¨Ãi§¢o§Ãٻ溺»Ãa¾¨¹¹Õée»š¹¨Ãi§eÊÁo·u¡æ
¡Ã³Õ.. æÊ´§Ç‹ÒÃٻ溺¹Õé “e»š¹Êa¨¹iÃa¹´Ã”
89. เพิ่มเติม บทที่ 1 ตรรกศาสตร์เบื้องต้น
89
μÇaoÂÒ‹§ ¾i¨ÒóҤ‹Ò¤ÇÒÁ¨Ãi§æμ‹Åa¡Ã³Õ¢o§Ã»Ù溺
»Ãa¾¨¹ (r ∧ p) ∨ (p→r) o´ÂÊÌҧμÒÃÒ§¤‹Ò¤ÇÒÁ¨Ãi§
´a§¹Õé
p r (r ∧ p) ∨ (p→r)
T T T
T F F
F T T
F F T
¾ºÇ‹ÒÃٻ溺»Ãa¾¨¹¹ÕéÁաóշÕèãËŒ¤‹Òe»š¹e·ç¨ä´Œ´ŒÇ (¤ืo
eÁืèo p e»š¹¨Ãi§æÅa r e»š¹e·ç¨) æÊ´§Ç‹ÒÃٻ溺¹Õé “äÁ‹e»š¹
Êa¨¹iÃa¹´Ã”
• ¡ÒÃμÃǨÊoºÇ‹Òe»š¹Êa¨¹iÃa¹´ÃËÃืoäÁ‹ ÊÒÁÒö㪌
“Çi¸Õ¾ÂÒÂÒÁ·íÒãËŒe»š¹e·ç¨” ¤ืoËÒ¡¾ÂÒÂÒÁ·íÒãËŒÃٻ溺
¹aé¹e»š¹e·ç¨äÁ‹ä´ŒeÅ Ãٻ溺¹a鹡ç¨ae»š¹Êa¨¹iÃa¹´Ã
(æ싶ŒÒ·íÒe»š¹e·ç¨ä´ŒæÁŒe¾Õ§¡Ã³Õe´ÕÂÇ Ãٻ溺¹aé¹Â‹oÁäÁ‹ãª‹
Êa¨¹iÃa¹´Ã)
μÇaoÂÒ‹§ ¨Ò¡ã¹μaÇo‹ҧe´iÁ ¡Òþi¨ÒóÒÇ‹ÒûÙ溺
»Ãa¾¨¹ [(p→q) ∧(q→r)]→(p→r) æÅa
(r ∨ p)→(p→r) e»š¹Êa¨¹iÃa¹´ÃËÃืoäÁ‹ o´ÂäÁ‹μŒo§ÊÌҧ
μÒÃÒ§¤‹Ò¤ÇÒÁ¨Ãi§ ÊÒÁÒö·íÒä´Œ´a§¹Õé..
(ãËŒ¾ÂÒÂÒÁËҡóշÕè·íÒãËŒ¤‹Ò¤ÇÒÁ¨Ãi§e»š¹e·ç¨ãˌ䴌)
90. สรุปคณิตศาสตร์ ม.4 เทอม 1
90
..Ãٻ溺 [(p→q) ∧(q→r)]→(p→r)
F
T T F
T F
T T F F
¾ºÇ‹Ò¤‹Ò q ¨a¢a´æÂŒ§¡a¹ äÁ‹Å§μaÇ æÊ´§Ç‹ÒäÁ‹ÊÒÁÒö·íÒ
ãËŒe»š¹e·ç¨ä´ŒeÅÂæÁŒæμ‹¡Ã³Õe´ÕÂÇ.. ´a§¹aé¹Ãٻ溺¹Õé “e»š¹Êa¨
¹iÃa¹´Ã”
..Ãٻ溺 (r ∧ p) ∨ (p→r)
F
F F
F T T F
¾ºÇ‹ÒÊÒÁÒö·íÒãËŒe»š¹e·ç¨ä´ŒÅ§μaǾo´Õ.. ´a§¹aé¹Ãٻ溺¹Õé
“äÁ‹e»š¹Êa¨¹iÃa¹´Ã”
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..æμ‹ÊíÒËÃaºÃٻ溺 ,↔+ ¤ÇÃ㪌ËÅa¡¡ÒÃÇ‹Ò ¨ae»š¹
Êa¨¹iÃa¹´ÃeÁืèo , ≡ + e·‹Ò¹aé¹
μÇaoÂÒ‹§ ûÙ溺»Ãa¾¨¹ ~(p→~q)↔(p ∧ q) æÅa
[p→(q→r)]↔[(p→q)→r] e»š¹Êa¨¹iÃa¹´ÃËÃืoäÁ ‹
92. สรุปคณิตศาสตร์ ม.4 เทอม 1
92
10. Çi¸ÕμÃǨÊoº¤ÇÒÁÊÁeËμuÊÁ¼Å ¢o§¡ÒÃoŒÒ§eËμu¼Å
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μ‹oeÁืèoÃٻ溺 “ ∧ ∧ ∧ ∧ → 1 2 3 n (p p p ... p ) ¼Å ” e»š¹
Êa¨¹iÃa¹´Ã (¹a蹤ืo ¨aäÁ‹ÊÁeËμuÊÁ¼Åe¾Õ§¡Ã³Õe´ÕÂÇ
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eËμu 1. ¶ŒÒÊÁªÒ¢Âa¹æÅŒÇe¢Ò¨aÊoºä´Œ
2. ¶ŒÒÊÁªÒÂäÁ‹¢Âa¹æŌǾ‹oæÁ‹¨aeÊÕÂã¨
3. ÊÁªÒÂÊoºäÁ‹ä´Œ
¼Å ¾‹oæÁ‹eÊÕÂã¨
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ɳ䴌´a§¹Õé
eËμu 1. p →q
2. ~ p →r
3. ~ q
¼Å r
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μÒÁÃٻ溺 [(p→ q) ∧ (~ p →r) ∧ (~ q)]→r
F
T T T F
T T T F F
