The reason of Boolean algebra is to simplify the design of a digital circuit that designed to performs a function, this helped to minimize the number of gates or to minimize the time of the function. We already discussed the Booleans laws and some theorems for this purpose, I will give some examples. | |||||
Simplification (Using Boolean Laws) | |||||
| Example 1 | Example 2 | Example 3 | |||
X.Y + X.Z + X.Y.Z =XY +XYZ + XZ =XY(1+Z) +XZ =XY + X =X(Y+Z) | [1+Z=1] | X.Y(X.Y+YZ) = (XY). (XY) + (XY).(YZ) = XY + XYZ = XY(1+Z) = XY.1 = XY | [Distributive Law] [ X.X = X] [ 1 + X = 1] | = X’YZ + XY’Z’ + XY’Z + XYZ' +XYZ = X’YZ + XY’ + XY = X’YZ + X = X + YZ | [xy’ + xy = x] |
Proof (Using Boolean Laws) | |||||
| Example 1 | Example 2 | Example 3 | |||
.XY + YZ + Y'Z = XY + Z =XY + YZ + Y'Z = XY + Z(Y + Y') = XY + 1.Z = XY + Z | [ AB +AC = A(B+C)] [ A + A' = 1] [ 1.A = A] | (X+Y)(X+Z) = X + YZ =XX + XZ + XY + YZ =X+XZ+XY+YZ =X.1 + XY + XZ + YZ =X(1+Y) + XZ + YZ =X + XZ + YZ =X(1+Z) + YZ =X+YZ | [A.(B+C)=AB + AC] [A.A = 1] [X.1 = X] [1 + A = 1] [1 + A = 1] | (X+Y)(Y+Z)(X+Z)=XY+XZ+YZ (XY + XZ + YY + YZ) (X+Z) (XY + XZ + Y (1 + Z)(X + Z) (XY + XZ + Y)(X + Z) XXY + XYZ + XXZ + XZZ + XY + YZ XY(1 + Z) + XZ + XZ + XY + YZ XY + XZ + YZ | [A(B+C) = AB + AC] [ 1 + A = 1] [ A. A = A] [ A. A = A] [ 1 + A = 1] |
Proof (Using Boolean Laws) | |||||||||||||||||||
| XY + YZ + Y'Z = XY + Z | (X+Y)(X+Z) = X + YZ | ||||||||||||||||||
| X | Y | Z | Y' | XY | YZ | Y'Z | XY+YZ+Y'Z | XY+Z | X | Y | Z | X+Y | X+Z | YZ | (X+Y)(X+Z) | X+YZ | |||
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||
| 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | |||
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | |||
| 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |||
| 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | |||
| 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | |||
| 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | |||
| 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |||
Propositional Logic | |||||||
Propositional Logic are concerned with propositional (or sentential) operators which may be applied to one or more propositions giving new propositions. Propositional logic is a propositional logic whose interpretation limits the truth values of its propositions to two, usually true and false that denoted by 1 and 0. | |||||||
1. NOT Operation (Inverter) | 2. OR Operation (Disjunction) | 3. AND Operation (Conjunction) | |||||
| ~p ≡ "Today is not Sunday" | Today is Sunday Or a Holiday (p ۷ q) | Today is Sunday and a Holiday (p ۸ q) | |||||
| Truth Table | Truth Table | Truth Table | |||||
p | ~p | p | q | p + q | p | q | p . q |
T | F | T | T | T | T | T | T |
| F | T | T | F | T | T | F | F |
| F | T | T | F | T | F | ||
| F | F | F | F | F | F | ||
| 4. IF - THEN (Conditional) Operation | 4. IF AND ONLY IF (Bi-conditional) Operation | |||||||||
“If today is Sunday, then today is my Birthday” | “If and only if today is Sunday, then today is my Birthday” | |||||||||
| p ≡ If today is Sunday q ≡ then today is my Birthday If p THEN q p → q | p ≡ If and only if today is Sunday q ≡ then today is my Birthday If and only if p THEN q p ↔ q | |||||||||
| Therefore p → q ≡ p' + q | Therefore p ↔ q ≡ p . q + p' . q' | |||||||||
| p | p' | q | p→q [ p' + q ] | p | p' | q | q' | p . q | p' . q' | p . q + p' . q' |
| T | F | T | T | T | F | T | F | T | F | T |
| T | F | F | F | T | F | F | T | F | T | F |
| F | T | T | T | F | T | T | F | F | T | F |
| F | T | F | T | F | T | F | T | F | T | T |
Tautology is a statement that is always true p ۷ ~q will always be True (Negation Law) Contradiction is a statement that is always false p ۸ ~p will always be False (Negation Law) | A logical equivalence means that the two sides always have the same truth values p → q and ~q → ~p is Logically equivalence | ||||
| p | p ۷ ~q | p ۸ ~p | p | q | p ≡ q |
| T | T | F | F | F | T |
| F | T | F | F | T | T |
| T | T | F | |||
| T | T | T | |||
Inversion: "If something is not a bird, then it is not an animal." Unlike the contrapositive, the inversion's truth value is not dependent upon whether the original proposition is true, as evidenced here. The inverse, here, is clearly not true.
Conversion: "If something is an animal, then it is a bird." The conversion is actually the contrapositive of the inversion and always has the same truth value as the inversion, which is not necessarily the same as that of the original proposition.
Contradiction: "There exists a bird that is not an animal. " If the contradiction is true, then the original proposition and, by extension, the contrapositive are untrue. Here, of course, the contradiction is untrue.
!!!Coming back with Maurice Karnaugh, I mean more with Boolean
Algebra and Karnaugh Map!!!
Algebra and Karnaugh Map!!!
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