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Lec5_BooleanAlgebra_11_updated

# Lec5_BooleanAlgebra_11_updated - Lecture 5 Boolean Algebra...

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1 Lecture 5 Boolean Algebra and Digital Logic

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2 Boolean Algebra Boolean algebra: The theoretical basis used in the analysis and design of electronic digital circuits. It provides the operations and the rules for working with the set {0,1} – electronic and optical switches can be studied using this set and the rules. The operation of a circuit is defined by a Boolean function that specified the value of an output for each set of inputs. The first step in constructing a circuit is to represent its Boolean function by an expression built up using the basic operations of Boolean algebra . Boolean algebra makes use of variables and operations. A variable take on that 1 : TRUE 0 : FALSE For example D = B + ( Ā · C) If B is 1 or if both A = 0 and C =1, D is equal to 1 Basic Operations ( Notation) AND : A AND B = A · B = A ^ B OR : A OR B = A + B = A v B NOT : NOT A = Ā = A
3 Gate -Level Concepts Level Concepts Switch circuits All switches have binary input/output signals and two states: open and closed (corresponding to 1 and 0). Logic gates: constructed using switch circuits NOT (Inverter) AND and OR gates NAND and NOR gates EXCLUSIVE-OR a nd NOR-gates All operations including arithmetic and logic are implemented along with the above basic logical functions and their combinational circuitry in ALU. Three main ways to represent logic gate 1. symbolic representation 2. logical expression: z = f ( X ) = f ( x1, x2, ..., xn ) 3. truth table: n variables 2 n different cases

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4 Gate -Level Concepts Level Concepts 1-bit half adder Example: half adder adds two binary digits to form Sum and Carry A B 0 0 1 1 0 1 0 1 Sum Carry 0 1 1 0 0 0 0 1 A + B ? C ? S
5 Boolean Algebra 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 F C B A Logic Circuit for Y = A’ B’ C’ + A’ B C + A B’ C’ Suppose we have an unknown digital circuit, represented by the given block box and conditions for high output. All we know is which terminals are inputs, which are output, and how to connect power. Given only that information, we can find the Boolean expression of the output. When A AND B AND C are all LOW, OR When A is LOW AND B AND C are HIGH, OR When A is HIGH AND B AND C are all LOW.

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6 Boolean Algebra Practical Example: Design a majority vote lighting circuit for three input switches which will turn on a light only if a majority of input are HIGH.
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Lec5_BooleanAlgebra_11_updated - Lecture 5 Boolean Algebra...

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