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INTRODUCTION

- “BOOLEAN ALGEBRA” also known as switching algebra and logical algebra was published by GEORGE BOOLE
- It’s a convenient and systematic way of expressing and analyzing the operation of logic circuits
- A Boolean Algebra is a closed algebraic system containing a set of two or more elements and two binary operators + (OR) and · (AND)
- VARIABLE – Symbol used to represent a logical quantity.
- COMPLEMENT – Inverse of a variable and is indicated by a bar over the variable
- LITERAL – Variable or the complement of a variable

The relationship between a single variable X, its complement X, and the binary constants 0 and 1

SALIENTS X

- OPERATOR PRECEDENCE
- PARENTHESES
- NOT
- AND
- OR

- TWO EXPRESSIONS ARE EQUIVALENT IF THEY HAVE IDENTICAL VALUES FOR ALL THE COMBINATIONS OF THE VALUES OF THEIR VARIABLES.
- ANY ALGEBRAIC EQUALITY DERIVED FROM THE AXIOMS OF BOOLEAN ALGEBRA REMAINS TRUE WHEN THE OPERATORS OR & AND ARE INTERCHANGED AND THE IDENTITY ELEMENTS 0 AND 1 ARE INTERCHANGED. THIS PROPERTY IS CALLED THE DUALITY PRINCIPLE. FOR EXAMPLE,
- X + 1 = 1X * 0 = 0 (DUAL)
- IF AN EQUATION IS VALID, THEN ITS DUAL IS ALSO VALID.

- PRODUCT TERM BY ANDING TWO OR MORE VARIABLES
- SUM TERM BY ORING TWO OR MORE TERMS

BOOLEAN ADDITION X

- Boolean addition is equivalent to the OR operation
- A sum term is produced by an OR operation with no AND ops involved.
- A sum term is equal to 1 when one or more of the literals in the term are 1.
- A sum term is equal to 0 only if each of the literals is 0.

1+0 = 1

0+1 = 1

1+1 = 1

0+0 = 0

BOOLEAN MULTIPLICATION X

- Boolean multiplication is equivalent to the AND operation
- A product term is produced by an AND operation with no OR ops involved.
- A product term is equal to 1 only if each of the literals in the term is 1.
- A product term is equal to 0 when one or more of the literals are 0.

0·1 = 0

1·0 = 0

1·1 = 1

0·0 = 0

CONSTRUCTING A TRUTH TABLE FOR A LOGIC CIRCUIT X

A(B+CD)=1

- When A=1 and B=1 regardless of the values of C and D

- When A=1 and C=1 and D=1 regardless of the value of B

EXERCISE X

INTRODUCTION X

DIGITAL SYSTEMS ARE CONSTRUCTED BY USING LOGIC GATES. THESE GATES ARE THE AND, OR, NOT, NAND, NOR, EXOR AND EXNOR GATES

- The most elementary logic elements are called GATES. They are used as the basic building blocks for constructing more complex logic devices.
- Electronic gates require a power supply.
- Gate INPUTS are driven by voltages having two nominal values, e.g. 0V and 5V representing logic 0 and logic 1 respectively.
- The OUTPUT of a gate provides two nominal values of voltage only, e.g. 0V and 5V representing logic 0 and logic 1 respectively. In general, there is only one output to a logic gate except in some special cases.
- There is always a time delay between an input being applied and the output responding.

AND GATE X

THE AND GATE IS AN ELECTRONIC CIRCUIT THAT GIVES A HIGH OUTPUT (1) ONLY IF ALL ITS INPUTS ARE HIGH. A DOT (.) IS USED TO SHOW THE AND OPERATION I.E. A.B. BEAR IN MIND THAT THIS DOT IS SOMETIMES OMITTED I.E. AB

OR GATE X

THE OR GATE IS AN ELECTRONIC CIRCUIT THAT GIVES A HIGH OUTPUT (1) IF ONE OR MORE OF ITS INPUTS ARE HIGH. A PLUS (+) IS USED TO SHOW THE OR OPERATION.

NOT GATE X

THE NOT GATE IS AN ELECTRONIC CIRCUIT THAT PRODUCES AN INVERTED VERSION OF THE INPUT AT ITS OUTPUT. IT IS ALSO KNOWN AS AN INVERTER. IF THE INPUT VARIABLE IS A, THE INVERTED OUTPUT IS KNOWN AS NOT A. THIS IS ALSO SHOWN AS A' OR A WITH A BAR OVER THE TOP, AS SHOWN AT THE OUTPUTS. THE NAND LOGIC GATE CAN BE CONFIGURED TO PRODUCE A NOT GATE. IT CAN ALSO BE DONE USING NOR LOGIC GATES IN THE SAME WAY.

THIS IS A NOT-AND GATE WHICH IS EQUAL TO AN AND GATE FOLLOWED BY A NOT GATE. THE OUTPUTS OF ALL NAND GATES ARE HIGH IF ANY OF THE INPUTS ARE LOW. THE SYMBOL IS AN AND GATE WITH A SMALL CIRCLE ON THE OUTPUT. THE SMALL CIRCLE REPRESENTS INVERSION.

THIS IS A NOT-OR GATE WHICH IS EQUAL TO AN OR GATE FOLLOWED BY A NOT GATE. THE OUTPUTS OF ALL NOR GATES ARE LOW IF ANY OF THE INPUTS ARE HIGH.

THE SYMBOL IS AN OR GATE WITH A SMALL CIRCLE ON THE OUTPUT. THE SMALL CIRCLE REPRESENTS INVERSION.

THE EXCLUSIVE-OR GATE IS A CIRCUIT WHICH WILL GIVE A HIGH OUTPUT IF EITHER, BUT NOT BOTH, OF ITS TWO INPUTS ARE HIGH. AN ENCIRCLED PLUS SIGN IS USED TO SHOW THE EOR OPERATION.

THE EXCLUSIVE-NOR GATE CIRCUIT DOES THE OPPOSITE TO THE EOR GATE. IT WILL GIVE A LOW OUTPUT IF EITHER, BUT NOT BOTH, OF ITS TWO INPUTS ARE HIGH. THE SYMBOL IS AN EXOR GATE WITH A SMALL CIRCLE ON THE OUTPUT. THE SMALL CIRCLE REPRESENTS INVERSION.

SUMMARY X

- AND gates: The output is true if and only if all inputs are true
- OR gates: The output is true if and only if any one of the inputs is true
- NAND gates: The output is false if and only if all inputs are true
- NOR gates: The output is false if and only if any one of the inputs is true
- EXCLUSIVE OR gates: The output is true if and only if the number of true inputs is odd
- EQUIVALENCE gates: The output is true if and only if the number of true inputs is even

THE XNAND AND NOR GATES ARE CALLED UNIVERSAL FUNCTIONS SINCE WITH EITHER ONE THE AND & OR & NOT FUNCTIONS CAN BE GENERATED.

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