Re: [問題] 基本定理和卡諾圖化簡問題

[guessi (沒)]

※ 引述《rx10a (奶油濺兔)》之銘言：
: 以下有幾個問題在自修時讀不懂煩請朋友們解說 謝謝!
: Q1:(A) XY+YZ+X'Z=XY+X'Z  (B)(X+Y)(Y+Z)(X'+Z)=(X+Y)(X'+Z)
: Q2:卡諾圖化簡請麻煩幫我舉例說明 謝謝 (光看書本化簡步驟不懂)
: Q3:(A) X(X+Y)=X+XY=X(1+Y)=X     (B) (X+Y)(X+Z)=X+XZ+XY+YZ=X(1+Z+Y)+YZ=X+YZ


若有錯誤，煩請指教更正... 3Q


Q1-A:  XY + YZ + X'Z = XY + X'Z

卡諾圖(左式)                             卡諾圖(右式)

╲XY                                     ╲XY
Z ╲  00  01  11  10                     Z ╲  00  01  11  10
    ┌─┬─┬─┬─┐                       ┌─┬─┬─┬─┐
0   │  │  │１│  │                   0   │  │  │１│  │
    ├─┼─┼─┼─┤                       ├─┼─┼─┼─┤
1   │１│１│１│  │    XY             1   │１│１│１│  │   XY
    └─┴─┴─┴─┘                       └─┴─┴─┴─┘

╲XY                      +              ╲XY                     +
Z ╲  00  01  11  10                     Z ╲  00  01  11  10
    ┌─┬─┬─┬─┐                       ┌─┬─┬─┬─┐
0   │  │  │１│  │         ===>>     0   │  │  │１│  │
    ├─┼─┼─┼─┤         ===>>         ├─┼─┼─┼─┤
1   │１│１│１│  │   YZ              1   │１│１│１│  │   X'Z
    └─┴─┴─┴─┘                       └─┴─┴─┴─┘

╲XY                     +
Z ╲  00  01  11  10
    ┌─┬─┬─┬─┐
0   │  │  │１│  │                   看圖應該就懂了... QQ
    ├─┼─┼─┼─┤
1   │１│１│１│  │   X'Z
    └─┴─┴─┴─┘



Q1-B: (X+Y)(Y+Z)(X'+Z)=(X+Y)(X'+Z)

  (X+Y)(Y+Z)(X'+Z)
= (X+Y)(X'Y + YZ + X'Z + Z)
= (X+Y)(X'Y + Z(X' + Y + 1))
= (X+Y)(X'Y + Z)
= XX'Y + XZ + X'Y + YZ
= XX' + XZ + X'Y + YZ
= X'(X+Y) + Z(X+Y)
= (X+Y)(Z'+Z) ........#


Q2: 請參考Q1-A


Q3-A: X(X+Y)=X+XY=X(1+Y)=X

  X(X+Y)
= XX + XY
= X + XY
= X．1 + X．Y
= X(1+Y)
= X(1)
= X  ..........#

Q3-B: (X+Y)(X+Z) = X+XZ+XY+YZ = X(1+Z+Y)+YZ=X+YZ

  (X+Y)(X+Z)           ↑            ↑
= XX+XZ+YX+YZ          │            │
= X + XZ + XY + YZ   ─┘            │
= X．1 + X．Z + X．Y + Y．Z          │
= X(1 + Z + Y) + YZ  ────────┘
= X + YZ .....#






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