Sets Key Part 001

 
 
   1 1.

 P(A)     P(B)  A  B     n(X)    !"ก  X  A  B $  % &'ก ()*  "( A ⊆ B  ( A ∪ B = B + x ∈ A ∪ B )( x ∈ A +*, x ∈ B  $, x ∈ B  ,ก x ∈ A  ( {x} ∈ P(A) ( x ∈ B  ( {x} ∈ P(B) )( P(A) ⊆ P(B)  ( ") - .ก
 ก.

2.

ก+  () A = {x ∈ R | x2  2x + 1 = 0}, B = {x ∈ R | x2  3x + 2 = 0}, C = { x ∈ R | x2  5x + 6 = 0} +$  % &'(   ก( , + !"ก ( ก(  A = {1}, B = {1, 2}, C = {2, 3} +0 )!( A ⊆ B  C
 .

3.

ก+ +  A, B, C    ) (!(   '  A ⊆ B  B ⊆ C    !"ก   $( ก 1   ,ก A ⊆ B  B ⊆ C )( A ∩ B = φ, A ∩ C = φ, B ∩ C = φ +*,ก0$, -  A, B, C  " *(ก ก n(A ∪ B ∪ C) = n(A) + n(B) + n(C)  n(A ∩ B)  n(A ∩ C)  n(B ∩ C) + n(A ∩ B ∩ C) = n(A) + n(B) + n(C) -----(*)  ,ก- 3   " *(ก ก ก* (*) 7)(  !"ก.  - 3   $((ก   !"ก- 3   กก )(   !"ก ก 1
 ก.

4.

 A   $  ก* x4  4x3  x2 + 16x  12 = 0, B = {1, 2, 3} +(  A  B  $  % &'ก ()*  ' -  *$ *.)7 2 *, $, *, ก *,*   *" ()+ !"ก  A  ,ก, ) A .( *. ก*%+1  ก*  ก*+

!"ก A 7)ก*ก ก*+$( x   -!:;< =;+,

ก+ + P(x) =  ,ก P(2) =  ( x  2 +* P(x) P(x) = = = ) x =  - A = +0 )!( B ⊆ A

x4  4x3  x2 + 16x  12 16  32  4 + 32  12 = 0  )( (x  2)(x3  2x2  5x + 6) (x  2)(x + 2)(x2  4x + 3) (x  2)(x + 2)(x  3)(x  1) = 0  2, 1, 2, 3 { 2, 1, 2, 3}
 .

5.

.ก , A  B    ) (!( (   ก. ) (.ก %* A ∩ B = {} ก0( , A ⊆ B ) (!( A ∪ B = {} ก0( , A ⊆ B  . .ก %* A ∪ (B ∩ C) = (A ∪ B) ∩ ( A ∪ C)  ก< *'ก  $. ) (.ก %* A  B = {x | x ∈ A  x ∉ B} ) (!( A  B = {x | x ∈ A  x ∈ B}  . ) (.ก %* A ∩ B = {x | x ∈ A  x ∈ B} ) (!( A ∩ B = {x | x ∉ A  x ∈ B}
 .

6.

" ก* " ก* ∗ () A ∗ B = (A ∩ B′)  (B ∩ A′)  U   กB%  %&', A, B, C    ) (  () (  ( *( ก  n(X) $,   !"ก  X %"*C(() -ก(.ก  %"*C ก. ก' A ∗ B = (A ∩ B′)  (B ∩ A′) = (A  B)  (B  A) = AB = A  B ∗ A = (B ∩ A′)  (A ∩ B′) = (B  A)  (A  B) = BA = B  ( ก. ) (.ก

()%"*C . ก (A ∗ B) ∗ C = =  A ∗ (B ∗ C) = =  ( . ) (.ก ()%"*C $. ก n(A ∗ B) = = =  n(B ∗ A) = = =  ( $. ) (.ก 7.

[(A ∩ B′)  (B ∩ A′)] ∗ C A∗C A ∗ (B  C) A∗B

n(A  B) n(A)  n(A ∩ B) n(A) ( ,ก n(A ∩ B) = 0) n(B  A) n(B)  n(A ∩ B) n(B) ( ,ก n(A ∩ B) = 0)   .

" ก* " ก* ∆  A ∆ B = {x  y | x ∈ A, y ∈ B  x > y}  , A = {2, 1, 0, 1, 2, 3}, B = { 1, 0, 1, 2, 3} () -ก(.ก  ก A ∆ B = {x  y | x ∈ A, y ∈ B  x > y} %"*C !"ก   A ∆ B  ,ก x ∈ A, y ∈ B  x > y +0 ( 0 > 1, 1 > 1, 1 > 0, 2 > 1, 2 > 0, 2 > 1, 3 > 1, 3 > 0, 3 > 1, 3 > 2 )( A ∆ B = {1, 2, 3, 4} %"*C,ก ก. *+(  {x | x5  6x4 + 12x3  12x2 + 11x  6 = 0} *ก  !"ก)*  + P(x) = x5  6x4 + 12x3  12x2 + 11x  6 = 0 :;< =;+,)( P(1) = 0 )( P(x) = (x  1)(x4  5x3 + 7x2  5x + 6) !:;< =;+,ก$*-)( P(2) = 0  - P(x) = (x  1)(x  2)(x3  3x2 + x  3) = (x  1)(x  2)[x2(x  3) + (x  3)] = (x  1)(x  2)(x  3)(x2 + 1) = 0  - {x | x5  6x4 + 12x3  12x2 + 11x  6 = 0} = {1, 2, 3, i, i}  ( {1, 2, 3, 4}  {x | x5  6x4 + 12x3  12x2 + 11x  6 = 0}  $,  ก. ) (.ก

%"*C,ก . )( (A ∆ B) ∪ A = {2, 1, 0, 1, 2, 3, 4}  (A ∆ B) ∪ B = {1, 0, 1, 2, 3, 4}  ,ก n[(A ∆ B) ∪ A] > n[(A ∆ B) ∪ B]  ( (A ∆ B) ∪ A  (A ∆ B) ∪ B  -  . ก() (.ก ()%"*C,ก $. ก (A ∆ B)  A = {4}  (A ∆ B)  B = {4} +0 ( (A ∆ B)  A = (A ∆ B)  B  ( $. ก(.ก 8.

  .

" ก* " ก* +! () A + B = {x + y | x ∈ A, y ∈ B  x + y ≠ 0} ก+ + A = {1, 0, 1}, B = {0, 1, 2}, C = {0, 1}  n(X) $,  !"ก  X () ก(.ก (%"*C,กก')  %"*C,ก ก. %"*C$( x + y  x ∈ A, y ∈ B  x + y ≠ 0 กก**    ก*C x = 1 ) (1, 0), (1, 1), (1, 2) *   ) 3 "& ก*C x = 0 ) (0, 0), (0, 1), (0, 2) *   ) 3 "& ก*C x = 1 ) (1, 0), (1, 1), (1, 2) *   ) 3 "& +0 ( % (1, 1)  (0, 0) ( - ) (*ก , )( x + y ≠ 0  (  A + B   !"ก$, {(1 + 0), (1 + 2), (0 + 1), (0 + 2), (1 + 0), (1 + 1), (1 + 2)} = {1, 1, 2, 3}  $, n(A + B) = 4 ≠ n(A) + n(B) = 6 %*F -  ก. ) (.ก ()%"*C . %"*C  ก  ก. )( n[(A + B) + C] = 5 ≠ n(A) + n(B) + n(C) = 8 %*F -  . G" ()%"*C $.  ,ก n(A + B) = 4 %*F -     -+   A + B (ก 24 = 16 %*F -  $. .ก   .

9.

 0ก 1  (A  B′) ∩ (C  D′) ∩ (E  F′) ∩ & ∩ (Y  Z′)  A ⊆ B ⊆ C ⊆ D & ⊆ Y ⊆ Z  A, B, C, &, Z   ) (( $,  () (%"*C,กก')  %"*C  (A  B′) ∩ (C  D′) ∩ (E  F′) ∩ & ∩ (Y  Z′) = A∩B∩C∩D∩&∩X∩Y∩Z  ) -+ $ (   !"ก(  1  (%*'ก+ , )(1ก   ) (() .(   + 7  " *' ก! ก  ('ก+ , )%" " ( A ⊆ B ⊆ C ⊆ D & ⊆ Y ⊆ Z  (  $, )-+  - %   0ก 1 $,  A     ก.

10. ก+ + P, Q     * "&"I   0 ก%+1  ก* n (n    0 ก)  Pn(x) = {a0, a1, a2, &, an}  Qn(x) = {b0, b1, b2, &, bn}  P ≠ Q  an, bn ≠ 0 " ก*  " ก* +! $, P + Q = {ci = ai + bi 1ก 0 ≤ i ≤ n | ai ∈ P, bi ∈ Q} () -ก(G"   ก.  . +0 )!(.ก ($1C  *%" . ')(( G 7ก*%" . ') %"*C $. ก a1 = a0 + a2  b1 = b0 + b2 )( a1 + b1 = (a0 + b0) + (a2 + b2)  $, c1 = c0 + c2 ) (  * *1)( 2 | c1 %*F -  $. ) (.ก   .