BOOLEAN ALGEBRA
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BOOLEAN
ALGEBRA
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Boolean Algebra
I will take an umbrella with me if it is raining or the weather forecast is bad
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Boolean Algebra
I will take an umbrella with me if it is raining or the weather forecast is bad
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Boolean Algebra
I will take an umbrella with me if it is raining or the weather forecast is bad
A Simple Proposition
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Boolean Algebra
I will take an umbrella with me if it is raining or the weather forecast is bad
Propositions may be TRUE or FALSE, are functions of other propositions, and connected by logical connections (AND, OR,
NOT)
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Boolean Algebra
I will take an umbrella with me if it is raining or the weather forecast is bad
Truth Table
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Boolean Algebra
George Boole, the mathematician, developed boolean algebra to simplify the handling of complex connectives.
Boolean Algebra uses ordinary algebraic notation, and 1 for True and 0 for False.
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Boolean Algebra
I will take an umbrella with me if it is raining or the weather forecast is bad
U = R + F
Boolean equation
Raining Bad Forecast Umbrella0 0 0
0 1 1
1 0 1
1 1 1
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Boolean Algebra
If I do not take the car then I will take an umbrella if it is raining or the weather forecast
is bad
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Boolean Algebra
If I do not take the car then I will take an umbrella if it is raining or the weather forecast
is bad
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Boolean Algebra
If I do not take the car then I will take an umbrella if it is raining or the weather forecast
is bad
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Boolean Algebra
If I do not take the car then I will take an umbrella if it is raining or the weather forecast
is bad
U = C’. (W + R)
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Boolean Algebra
U = C’. (W + R)
If I do not take the car then I will take an umbrella if it is raining or the weather forecast
is bad
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Boolean Identities Basic boolean identities.
X + 0 = X X * 1 = X X + 1 = 1
X * 0 = 0 X + X = X X * X = X
X + X = 1 X = XX * X = 0
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Basic Laws of Boolean Algebra The basic laws of boolean algebra are:
Commutative Law:X + Y = Y + X XY = YX
Associative Law:X+(Y+Z) = (X+Y)+Z X(YZ)=(XY)Z
Distributive Law:X(Y+Z) = XY + XZ X+YZ = (X+Y)(X+Z)
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Boolean Laws and Identities Reduce the boolean function F = X Y Z + X Y + X Y Z
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Boolean Laws and Identities Reduce the boolean function F = X Y Z + X Y + X Y Z
F = X Y (Z + Z) + X Y (Distributive and Commutative Laws)
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Boolean Laws and Identities Reduce the boolean function F = X Y Z + X Y + X Y Z
F = X Y (Z + Z) + X Y (Distributive and Commutative Laws)
F = X Y (Z + Z) + X Y (Identity) = 1
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Boolean Laws and Identities Reduce the boolean function F = X Y Z + X Y + X Y Z
F = X Y (Z + Z) + X Y (Distributive and Commutative Laws)
F = X Y (Z + Z) + X Y (Identity) = 1F = X Y + X Y
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Boolean Laws and Identities Reduce the boolean function F = X Y Z + X Y + X Y Z
F = X Y (Z + Z) + X Y (Distributive and Commutative Laws)
F = X Y (Z + Z) + X Y (Identity) = 1F = X Y + X Y
F = (X + X) Y (Distributive Law)
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Boolean Laws and Identities Reduce the boolean function F = X Y Z + X Y + X Y Z
F = X Y (Z + Z) + X Y (Distributive and Commutative Laws)
F = X Y (Z + Z) + X Y (Identity) = 1F = X Y + X Y
F = (X + X) Y (Distributive Law)
F = (X + X) Y (Identity) = 1
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Boolean Laws and Identities Reduce the boolean function F = X Y Z + X Y + X Y Z
F = X Y (Z + Z) + X Y (Distributive and Commutative Laws)
F = X Y (Z + Z) + X Y (Identity) = 1F = X Y + X Y
F = (X + X) Y (Distributive Law)
F = (X + X) Y (Identity) = 1F = Y
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Advanced Boolean Laws Here are some advanced laws of boolean algebra that
can be directly applied to the reduction of boolean functions.
X + X Y = XX Y + X Y = XX + X Y = X + YX (X + Y) = X(X + Y)(X + Y) = XX (X + Y) = X
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