Докажите, что:
1) $\overset{=}{A} = A$;
2) A ∧ A = A;
3) A ∨ B = B ∨ A;
4) A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C);
5) $\overline{A ∨ B} = \overline{A} ∧ \overline{B}$;
6) $(A ⇒ B) = \overline{B} ⇒ \overline{A}$.
$\overset{=}{A} = A$
| A | $\overset{-}{A}$ | $\overset{=}{A}$ |
|---|---|---|
| 0 | 1 | 0 |
| 1 | 0 | 1 |
A ∧ A = A
| A | A | A ∧ A |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 1 | 1 |
A ∨ B = B ∨ A
| A | B | A ∨ B | B ∨ A |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 |
A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)
| A | B | C | B ∧ C | A ∨ (B ∧ C) |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 |
| A | B | C | A ∨ B | A ∨ C | (A ∨ B) ∧ (A ∨ C) |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 1 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 |
$\overline{A ∨ B} = \overline{A} ∧ \overline{B}$
| A | B | A ∨ B | $\overline{A ∨ B}$ |
|---|---|---|---|
| 1 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| A | B | $\overline{A}$ | $\overline{B}$ | $\overline{A} ∧ \overline{B}$ |
|---|---|---|---|---|
| 1 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 |
$(A ⇒ B) = \overline{B} ⇒ \overline{A}$
| A | B | A ⇒ B |
|---|---|---|
| 1 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 0 | 1 |
| A | B | $\overline{A}$ | $\overline{B}$ | $\overline{B} ⇒ \overline{A}$ |
|---|---|---|---|---|
| 1 | 1 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 1 |
Пожаулйста, оцените решение
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