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Консультация № 191986
04.12.2017, 07:56
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Сколько существует различных наборов значений логических переменных x1,x2,x3,x4,x5,x6,x7,x8,x9,x10, которые удовлетворяют всем перечисленным ниже условиям?
x1 /\ (x2 → x3) \/ ¬x1 /\ x4 = 1
x3 /\ (x4 → x5) \/ ¬x3 /\ x6 = 1
x5 /\ (x6 → x7) \/ ¬x5 /\ x8 = 1
x7 /\ (x8 → x9) \/ ¬x7 /\ x10 = 1
Приведите полное решение задачи с пояснениями.
Сколько существует различных наборов значений логических переменных x1,x2,x3,x4,x5,x6,x7,x8,x9,x10, которые удовлетворяют всем перечисленным ниже условиям?
x1 /\ (x2 → x3) \/ ¬x1 /\ x4 = 1
x3 /\ (x4 → x5) \/ ¬x3 /\ x6 = 1
x5 /\ (x6 → x7) \/ ¬x5 /\ x8 = 1
x7 /\ (x8 → x9) \/ ¬x7 /\ x10 = 1
Приведите полное решение задачи с пояснениями.
Обсуждение
13.12.2017, 18:39
общий
это ответ
Здравствуйте, IIISergeyIII!
Составим таблицу функции x[sub]1[/sub][$8743$](x[sub]2[/sub][$8594$]x[sub]3[/sub])[$8744$](-x[sub]1[/sub][$8743$]x[sub]4[/sub]):
Из неё видно, что первому условию удовлетворяют все наборы вида 0xx1xxxxxx, 10xxxxxxxx и 111xxxxxxx, где x - любое значение (всего 256+256+128=640 наборов). Аналогично, второму условию удовлетворяют все наборы вида xx0xx1xxxx, xx10xxxxxx и xx111xxxxx (полученные из предыдущих сдвигом вправо на две позиции). Тогда одновременно первому и второму условию будут удовлетворять следующие наборы: 0x01x1xxxx, 0x111xxxxx, 100xx1xxxx, 1010xxxxxx, 10111xxxxx, 1110xxxxxx и 11111xxxxx (всего 64+64+64+64+32+64+32=384 набора). Если учесть также наборы, удовлетворяющие третьему условию (xxxx0xx1xx, xxxx10xxxx и xxxx111xxx), то первым трём условиям будут удовлетворять следующие наборы: 0x0101x1xx, 0x01111xxx, 0x1110xxxx, 0x11111xxx, 100x01xxxx, 100x111xxx, 10100xx1xx, 101010xxxx, 1010111xxx, 101110xxxx, 1011111xxx, 11100xx1xx, 111010xxxx, 1110111xxx, 111110xxxx и 1111111xxx (16+16+32+16+32+16+16+16+8+16+8+16+16+8+16+8=256 наборов). Наконец, с учётом наборов, удовлетворяющих четвёртому условию (xxxxxx0xx1, xxxxxx10xx и xxxxxx111x), решением будет 0x010101x1, 0x0101111x, 0x011110xx, 0x0111111x, 0x11100xx1, 0x111010xx, 0x1110111x, 0x111110xx, 0x1111111x, 100x010xx1, 100x0110xx, 100x01111x, 100x1110xx, 100x11111x, 10100x01x1, 10100x111x, 1010100xx1, 10101010xx, 101010111x, 10101110xx, 101011111x, 1011100xx1, 10111010xx, 101110111x, 10111110xx, 101111111x, 11100x01x1, 11100x111x, 1110100xx1, 11101010xx, 111010111x, 11101110xx, 111011111x, 1111100xx1, 11111010xx, 111110111x, 11111110xx и 111111111x - всего 4+4+8+4+8+8+4+8+4+8+8+4+8+4+4+4+4+4+2+4+2+4+4+2+4+2+4+4+4+4+2+4+2+4+4+2+4+2=164 набора.
Составим таблицу функции x[sub]1[/sub][$8743$](x[sub]2[/sub][$8594$]x[sub]3[/sub])[$8744$](-x[sub]1[/sub][$8743$]x[sub]4[/sub]):
Из неё видно, что первому условию удовлетворяют все наборы вида 0xx1xxxxxx, 10xxxxxxxx и 111xxxxxxx, где x - любое значение (всего 256+256+128=640 наборов). Аналогично, второму условию удовлетворяют все наборы вида xx0xx1xxxx, xx10xxxxxx и xx111xxxxx (полученные из предыдущих сдвигом вправо на две позиции). Тогда одновременно первому и второму условию будут удовлетворять следующие наборы: 0x01x1xxxx, 0x111xxxxx, 100xx1xxxx, 1010xxxxxx, 10111xxxxx, 1110xxxxxx и 11111xxxxx (всего 64+64+64+64+32+64+32=384 набора). Если учесть также наборы, удовлетворяющие третьему условию (xxxx0xx1xx, xxxx10xxxx и xxxx111xxx), то первым трём условиям будут удовлетворять следующие наборы: 0x0101x1xx, 0x01111xxx, 0x1110xxxx, 0x11111xxx, 100x01xxxx, 100x111xxx, 10100xx1xx, 101010xxxx, 1010111xxx, 101110xxxx, 1011111xxx, 11100xx1xx, 111010xxxx, 1110111xxx, 111110xxxx и 1111111xxx (16+16+32+16+32+16+16+16+8+16+8+16+16+8+16+8=256 наборов). Наконец, с учётом наборов, удовлетворяющих четвёртому условию (xxxxxx0xx1, xxxxxx10xx и xxxxxx111x), решением будет 0x010101x1, 0x0101111x, 0x011110xx, 0x0111111x, 0x11100xx1, 0x111010xx, 0x1110111x, 0x111110xx, 0x1111111x, 100x010xx1, 100x0110xx, 100x01111x, 100x1110xx, 100x11111x, 10100x01x1, 10100x111x, 1010100xx1, 10101010xx, 101010111x, 10101110xx, 101011111x, 1011100xx1, 10111010xx, 101110111x, 10111110xx, 101111111x, 11100x01x1, 11100x111x, 1110100xx1, 11101010xx, 111010111x, 11101110xx, 111011111x, 1111100xx1, 11111010xx, 111110111x, 11111110xx и 111111111x - всего 4+4+8+4+8+8+4+8+4+8+8+4+8+4+4+4+4+4+2+4+2+4+4+2+4+2+4+4+4+4+2+4+2+4+4+2+4+2=164 набора.
5
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