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Logic Gates and Boolean Algebra

Logic Gates and Boolean Algebra
Boolean algebra is an important tool in describing, analyzing, designing, and implementing digital circuits.

Boolean Constants and Variables
Boolean algebra allows only two values; 0 and 1. Logic 0 can be: false, off, low, no, open switch. Logic 1 can be: true, on, high, yes, closed switch. Three basic logic operations: OR, AND, and NOT.

Truth Tables A truth table describes the relationship between the input and output of a logic circuit. The number of entries corresponds to the number of inputs. For example a 2 input table would have 22 = 4 entries. A 3 input table would have 23 = 8 entries.

Truth Tables Examples of truth tables with 2, 3, and 4 inputs.

Boolean algebra Used to design and analysis digital circuit
Invented by George Boole Uses variables and operations Variable may take value TRUE or FALSE Logical operations are AND,OR ,and NOT

Boolean Laws A. 0 = 0 A + 0 =A A . 1 =A A + 1 = 1 Ã + 1 = 1 A . A = A
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Boolean Laws A . B = B.A A + B = B +A A.(B.C)=(A.B).C A +(B+C)=(A+B)+C
A.(B+C)=A.B + A.C A+(B.C)=(A+B).(A+C) A.(A+B)=A

Boolean Laws A + AB =A A.(Ã + B) = AB A + ÃB= A + B Ã + AB = Ã + B

Duality Swap 0 & 1, AND & OR Principle of Duality (Metatheorem)
Result: Theorems still true Principle of Duality (Metatheorem) Any theorem or identity in switching algebra remains true if 0 and 1 are swapped and • and + are swapped throughout. Fully parenthesized before taking its duality

Duality Theorem Find dual of A + AB = A Step 1 = A .(A . B) = A
A. (A + B) = A is dual of A + AB =A

DeMorgan’s Theorem Equation: Equation:
The complement of two or more ANDed variables is equivalent to the OR of the complements of the individual variables Equation: The complement of two or more ORed variables is equivalent to the AND of the complements of the individual variables Equation:

A + AB = A(1 +B) DISTRIBUTIVE LAW = A∙1 (1+B)=1 = A A∙1 = A
Boolean Rules A + AB = A Proof: A + AB = A(1 +B) DISTRIBUTIVE LAW = A∙ (1+B)=1 = A A∙1 = A

(A + B)(A + C) = A + BC PROOF
(A + B)(A +C) = AA + AC +AB +BC DISTRIBUTIVE LAW = A + AC + AB + BC = A(1 + C) +AB + BC = A.1 + AB + BC = A(1 + B) + BC = A.1 + BC = A + BC

OR Operation With OR Gates
The Boolean expression for the OR operation is X = A + B This is read as “x equals A or B.” X = 1 when A = 1 or B = 1. Truth table and circuit symbol for a two input OR gate:

OR Operation With OR Gates
The OR operation is similar to addition but when A = 1 and B = 1, the OR operation produces = 1. In the Boolean expression x=1+1+1=1 We could say in English that x is true (1) when A is true (1) OR B is true (1) OR C is true (1).

AND Operations with AND gates
The Boolean expression for the AND operation is X = A • B This is read as “x equals A and B.” x = 1 when A = 1 and B = 1. Truth table and circuit symbol for a two input AND gate are shown. Notice the difference between OR and AND gates.

Operation With AND Gates
The AND operation is similar to multiplication. In the Boolean expression X = A • B • C X = 1 only when A = 1, B = 1, and C = 1.

NOT Operation The Boolean expression for the NOT operation is
This is read as: x equals NOT A, or x equals the inverse of A, or x equals the complement of A

NOT Operation Truth table, symbol, and sample waveform for the NOT circuit.

Describing Logic Circuits Algebraically
The three basic Boolean operations (OR, AND, NOT) can describe any logic circuit. If an expression contains both AND and OR gates the AND operation will be performed first, unless there is a parenthesis in the expression.

Describing Logic Circuits Algebraically
Examples of Boolean expressions for logic circuits:

Describing Logic Circuits Algebraically
The output of an inverter is equivalent to the input with a bar over it. Input A through an inverter equals A. Examples using inverters.

NOR Gates and NAND Gates
Combine basic AND, OR, and NOT operations. The NOR gate is an inverted OR gate. An inversion “bubble” is placed at the output of the OR gate. The Boolean expression is,

NOR Gates and NAND Gates
The NAND gate is an inverted AND gate. An inversion “bubble” is placed at the output of the AND gate. The Boolean expression is

NAND and NOR Gates NAND and NOR gates can greatly simplify circuit diagrams. Use these gates wherever you could use AND, OR, and NOT. A B AB 1 NAND A B AB 1 NOR

Universality of NAND and NOR Gates
NAND or NOR gates can be used to create the three basic logic expressions (OR, AND, and INVERT) This characteristic provides flexibility and is very useful in logic circuit design.

The Exclusive OR

The Exclusive NOR

XOR and XNOR Gates XOR is used to choose between two mutually exclusive inputs. Unlike OR, XOR is true only when one input or the other is true, not both. A B AB 1 XOR A B A B 1 XNOR

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