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