Boolean Algebra
Basic Engineering
Boolean Algebra and Logic Gates
F Hamer, M Lavelle & D McMullan The aim of this document is to provide a short, self assessment programme for students who wish to understand the basic techniques of logic gates.
c 2005 Email: chamer, mlavelle, dmcmullan@plymouth.ac.uk Last Revision Date: August 31, 2006 Version 1.0
Table of Contents
1. 2. 3. 4. 5. Logic Gates (Introduction) Truth Tables Basic Rules of Boolean Algebra Boolean Algebra Final Quiz Solutions to Exercises Solutions to Quizzes
The full range of these packages and some instructions, should they be required, can be obtained from our web page Mathematics Support Materials.
Section 1: Logic Gates (Introduction)
3
1. Logic Gates (Introduction)
The package Truth Tables and Boolean Algebra set out the basic principles of logic. Any Boolean algebra operation can be associated with an electronic circuit in which the inputs and outputs represent the statements of Boolean algebra. Although these circuits may be complex, they may all be constructed from three basic devices. These are the AND gate, the OR gate and the NOT gate. x y AND gate x·y x y OR gate x+y x NOT gate x
In the case of logic gates, a different notation is used: x ∧ y, the logical AND operation, is replaced by x · y, or xy. x ∨ y, the logical OR operation, is replaced by x + y. ¬x, the logical NEGATION operation, is replaced by x or x. The truth value TRUE is written as 1 (and corresponds to a high voltage), and FALSE is written as 0 (low voltage).
Section 2: Truth Tables
4
2. Truth Tables x y x·y
x 0 0 1 1 Summary
y x·y 0 0 1 0 0 0 1 1 of AND gate
x 0 0 1 1 Summary
y x+y 0 0 1 1 0 1 1 1 of OR gate
x y
x+y
x
x
x 0 1 Summary of
x 1 0 NOT gate
Section 3: Basic Rules of Boolean Algebra
5
3. Basic Rules of Boolean Algebra
The basic rules for simplifying and combining logic gates are called Boolean algebra in honour of George Boole (1815 – 1864) who was a
Boolean Algebra and Logic Gates
F Hamer, M Lavelle & D McMullan The aim of this document is to provide a short, self assessment programme for students who wish to understand the basic techniques of logic gates.
c 2005 Email: chamer, mlavelle, dmcmullan@plymouth.ac.uk Last Revision Date: August 31, 2006 Version 1.0
Table of Contents
1. 2. 3. 4. 5. Logic Gates (Introduction) Truth Tables Basic Rules of Boolean Algebra Boolean Algebra Final Quiz Solutions to Exercises Solutions to Quizzes
The full range of these packages and some instructions, should they be required, can be obtained from our web page Mathematics Support Materials.
Section 1: Logic Gates (Introduction)
3
1. Logic Gates (Introduction)
The package Truth Tables and Boolean Algebra set out the basic principles of logic. Any Boolean algebra operation can be associated with an electronic circuit in which the inputs and outputs represent the statements of Boolean algebra. Although these circuits may be complex, they may all be constructed from three basic devices. These are the AND gate, the OR gate and the NOT gate. x y AND gate x·y x y OR gate x+y x NOT gate x
In the case of logic gates, a different notation is used: x ∧ y, the logical AND operation, is replaced by x · y, or xy. x ∨ y, the logical OR operation, is replaced by x + y. ¬x, the logical NEGATION operation, is replaced by x or x. The truth value TRUE is written as 1 (and corresponds to a high voltage), and FALSE is written as 0 (low voltage).
Section 2: Truth Tables
4
2. Truth Tables x y x·y
x 0 0 1 1 Summary
y x·y 0 0 1 0 0 0 1 1 of AND gate
x 0 0 1 1 Summary
y x+y 0 0 1 1 0 1 1 1 of OR gate
x y
x+y
x
x
x 0 1 Summary of
x 1 0 NOT gate
Section 3: Basic Rules of Boolean Algebra
5
3. Basic Rules of Boolean Algebra
The basic rules for simplifying and combining logic gates are called Boolean algebra in honour of George Boole (1815 – 1864) who was a