Лекция дискрет 12
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Примеры применения «полезных соотношений»
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(3.2.3) |
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(x & y) ((z x) & ¬y) |
= (x & y) (z & ¬y) (x & ¬y) = |
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(3.2.3) |
(3.2.10) |
= ((x & y) (x & ¬y)) (z & ¬y) = |
(x & (y ¬y)) (z & ¬y) = |
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(3.2.7)
= (x & 1) (z & ¬y) = x (z & ¬y)
(3.3.5) |
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x (x & ((y & ¬z) (¬y & z))) = |
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(3.2.7) |
(3.2.7) |
= x (0 & ((y & ¬z) (¬y & z))) = |
x 0 = x |
3) Эквивалентные преобразования небулевских формул
(A)x y = (¬x & y) (x & ¬ y) = (x y) & (¬x ¬y)
(B)x y = (x & y) (¬x & ¬ y) = (x ¬y) & (¬x y)
(C) x y = ¬x & ¬y |
(D) |
x y = ¬x ¬y |
(E) 0 = x & ¬x (F) |
x y = ¬x y |
(G) 1 = x ¬x |
Пример 1
(F) |
(3.2.8) |
(x y) (( x&y) (x& y)) = (x y) ( x&y) (x& y) =
(3.3.2) |
(3.3.5) |
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= ( x& y) ( x&y) (x& y) |
= x (x& y) = x y |
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Пример 2 |
(F) |
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(F) |
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((x y) (x y)) & (( x y) (x y)) = |
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(A) |
(F) |
(A) |
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= ( (x y) x y) & ( ( x y) (x y)) =
(3.2.8)
= ( ((x y) & (¬x ¬y)) x y) &
3.3.1a + 3.3.1a
& ( x y (¬x & y) (x & ¬ y)) =
(3.2.8) (3.2.8)
=( (x y) (¬x ¬y) x y) & (x y) =
3.3.4+ 3.3.1a
=( ( x & y) (x & y) x y) & (x y) =
(3.2.10) |
(3.2.7) |
(3.2.7) |
= ( x y x)&(x y) = (1 y)&(x y) = 1&(x y) = x y