22c:19 discrete math boolean algebra & digital logic fall 2010 sukumar ghosh
TRANSCRIPT
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22C:19 Discrete MathBoolean Algebra & Digital Logic
Fall 2010Sukumar Ghosh
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Boolean Algebra
In 1938, Shannon showed how the basic rules of logicfirst given by George Boole in his 1854 publication The Laws of Thought, can be used to design circuitsthat enables computer hardware to perform variousarithmetic and logical operation. These rules form thebasis of Boolean Algebra. These rules involve twoSymbols 1 and 0 (synonymous with true and false) and the operators AND, OR, NOT.
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Boolean Algebra
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Basic Logic Gates
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Circuits to Equations
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Circuits to Equations
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Equations to Circuits
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Implementing XOR
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Boolean function
An n-variable Boolean function f : {0.1}n {0,1}⟶
It can be expressed as a truth table.
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Boolean Algebra
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Sum-of-product form
minterm
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Examples of simplification
A . B’ + A . B = A. (B’ + B) = A.1 = A
(A + B).(A + B’) = (A + B.B’) = A + 0 = A
In the reverse way, you can expand a expression,
Like X.Y = X.Y.1 = X.Y (Z + Z’) = X.Y.Z + X.Y.Z’
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NAND and NOR gates
Alternative notations: A | B = A NAND BA ↓ B = A NOR B
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Binary addition
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Half-adder
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Is half-adder enough?
Can we add two binary numbers using half adders only?
There is a problem
here
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Is half-adder enough?
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