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Axioms and Laws of Boolean Algebra
Identities0 + X = X | 0 · X = 0 | 1 + X = 1 | 1 · X = X | X + X = X | X · X = X |
NegationX + ~X = 1 | ~0 = 1 | ~1 = 0 | ~~X = X | X · ~X = 0 |
LawsCommunative Law | A · B = B · A | | | A + B = B + A | Associative Law | A · (B · C) = (A · B) · C | | | A + (B + C) = (A + B) +C | Distributive Law | A · (B + C) = A·B + A·C | | | A + B · C = (A+B)(A+C) |
| | De Morgan's Laws ~(X · Y) | = ~X + ~Y | ~(X + Y) | = ~X · ~Y | ~(X · Y · Z) | = ~X + ~Y + ~Z | ~(X + Y + Z) | = ~X · ~Y · ~Z |
TheoremsTheorem 1 X + X · Y = X | Theorem 2 X + ~X · Y = X + Y | Theorem 3 X · Y + ~X · Z + Y · Z = X · Y + ~X · Z | Theorem 4 X(X + Y) = X | Theorem 5 X(~X + Y) = X · Y | Theorem 6 (X + Y)(X + ~Y) = X | Theorem 7 (X + Y)(~X + Z) = X · Z + ~X · Y | Theorem 8 (X + Y)(~X + Z)(Y + Z) = (X + Y)(~X + Z) |
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