boolean logic
DESCRIPTION
detailed study on Boolean logic and its application by Cornelius saikiTRANSCRIPT
Boolean algebra (or Boolean logic) is a logical calculus of
truth values, developed by George Boole in the 1840s.
It resembles the algebra of real numbers but the numerical
operators
• Multiplication AB
• Addition A + B
• Negation ¬A
Been replaced by respective logical operator’s
• Conjunction (AND) AɅB
• Disjunction (OR) AVB
• Complement (NOT) ¬A
These are the basic Boolean operators and all other are built from these.
A Ʌ ( BV C)
Example:
Boolean Logic
Boolean Algebra Laws
Boolean algebra, like regular algebra, has certain rules.
• Associativity
• Distributivity
• Commutativity
Associativity, Commutativity and Distributivity only apply to the AND
and OR operators.
• ASSOCIATIVITY
Associativity is the property of algebra that the order of evaluation
of the terms is immaterial.
A V (B V C)=(A V B) V C
A Ʌ (B Ʌ C)=(A Ʌ B) Ʌ C
A + (B + C)=(A + B) + C
A . (B . C)=(A . B) . C
or
Distributivity
Distributivity is the property that an operator can be
applied to the terms within the brackets.
A Ʌ (B V C) =(A Ʌ B) V (A Ʌ C)
A. (B + C) =(A . B) + (A . C)
or
Commutativity
Commutativity is the property that order of application of
an operator is immaterial.
A Ʌ B = B Ʌ A
A V B = B V A or
A . B = B . A
A + B = B + A
De Morgan's Law
De Morgan's Law is a consequence of the fact that the NOT or
negation operator is not distributive.
¬(P Ʌ Q)= (¬P) V (¬Q)
¬(P V Q)= (¬P) Ʌ (¬Q)
or P . Q =P +Q
P + Q =P . Q
Rule 1: A +0=A Rule 7: A.A=A
Rule 2: A + 1=1 Rule 8: A.¬A=0
Rule 3: A.0=0 Rule 9: ¬ ¬A=0
Rule 4: A.1=A Rule 10: A+AB=A
Rule 5: A+A=A Rule 11: A+ ¬AB=A+B
Rule 6: A+¬A=1 Rule 12: (A+B)(A+C)=A+BC
These laws of Boolean results in a number of rules which then used
for Boolean algebra.
There are 12 basic rules of Boolean algebra
Examples
Simplify the following Expressions.
B Ʌ A Ʌ ¬A
Using rule 8. We get.
B Ʌ 0 = 0
A Ʌ ¬B V A Ʌ B
Taking out A.
A Ʌ (¬B V B )
This gives use A
1.
2
THEOREMS OF ALGEBRA AND TRUTH TABLE
There are several operators used in Boolean algebra (logics)
Conjunction (AND):
It is the closest of these three to its numerical counterpart, in
fact on 0 and 1 it is multiplication.
As a logical operation the conjunction of two propositions is
true when both propositions are true, and otherwise is false.
Boolean expression: X=A∩B
A B X=A∩B
F F F
F T F
T F F
T T T
A B C=A∩B
0 0 0
0 1 0
1 0 0
1 1 1
0 * 0 = 0
0 * 1 = 0
1 * 0 = 0
1 * 1 = 1
A B C X=A∩B∩C
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
Using three inputs
Boolean expression : X=A∩B∩C
0 * 0 * 0 = 0
0 * 0 * 1 = 0
0 * 1 * 0 = 0
0 * 1 * 1 = 0
1 * 0 * 0 = 0
1 * 0 * 1 = 0
1 * 1 * 0 = 0
1 * 1 * 1 = 1
Try other combinations of the input levels
Disjunction:
It operates almost like addition, with one
exception:
The disjunction of 1 and 1 is neither 2 nor 0 but
1.
Thus the disjunction of two propositions is false
when both propositions are false, and otherwise
is true.
Boolean expression: X=A∪B
A B X=A∪B
F F F
F T T
T F T
T T T
A B C=A∪B
0 0 0
0 1 1
1 0 1
1 1 1
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1
Using three inputs
Boolean expression : X=A∩B∩C
A B C X=A∪B∪C
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1
0 +0 + 0 = 0
0 + 0 + 1 = 0
0 + 1 + 0 = 0
0 + 1 + 1 = 0
1 + 0 + 0 = 0
1 + 0 + 1 = 0
1 + 1 + 0 = 0
1 + 1 + 1 = 1
Logical Negation (NOT):
Also called complement
Since we have only two numbers in Boolean algebra, If we know the value of
one, then we automatically know the value of the other.
Thus this number is the complement of the variable.
1 is the complement of 0
0 is the compliment of 1
Complement of A=¬A
A ¬A
0 1
1 0
A ¬A
F T
T F
NAND AND NOR LOGIC OPERATORS
NAND LOGIC OPERATORS
• This comes from two operators
NOT & AND
• It is made up from the compliment of their respective functions
A B X
F F T
F T T
T F T
T T F
A B X
0 0 1
0 1 1
1 0 1
1 1 0
• This comes from two operators NOT & AND
• It is made up from the compliment of their respective functions
NOR LOGIC OPERATORS
NAND AND NOR LOGIC OPERATORS
A B X=A∪B
F F F
F T T
T F T
T T T
A B C=A∪B
0 0 0
0 1 1
1 0 1
1 1 1
A B X
0 0 0
0 1 1
1 0 1
1 1 0
Exclusive OR & NOR OPERATORS (XOR & XNOR)
• Gives false if both inputs are true or both are
false .
• Gives true if we have different input levels
EXCLUSIVE OR (XOR)
It is the negative complement of XOR operator
Exclusive NOR (XNOR)
A B X
0 0 1
0 1 0
1 0 0
1 1 1
Boolean logic and Electronics Circuits
Boolean algebra is devised for dealing mathematically with philosophical
propositions which have ONLY TWO possible values: TRUE or FALSE,
Light ON or OFF.
SW1 Open >> Lamp is OFF
SW1 Closed >> Lamp is ON
The truth table for these conditions is stated below
SW1 Lamp
OPEN OFF
CLOSED ON
SW1 Lamp
0 0
1 1
Electronic Systems:
Analogue >> Continuous System
Digital >> Discrete System
Boolean logic and Electronics Circuits In Boolean algebra the TWO possible conditions can
be represented by the DIGITS
“0” and “1”.
Binary Digits – Bits.
Light ON = “1” = +5V = HIGH
Light OFF = “0” = 0V = LOW
Open = “0”, CLOSED = “1”
“AND” operation: Describes events which can
occur IF and only IF 2 or more other events are
TRUE
Consider the circuit below A B C
OPEN OPEN OFF
OPEN CLOSED OFF
CLOSED OPEN OPEN
CLOSED CLOSED ON
A B L
0 0 0
0 1 0
1 0 0
1 1 1
Lamp will light ONLY when the switches A and B are
CLOSED, i.e. A and B both “1”
NOTATION: C = A.B
C = AB Boolean Equation
The symbol for AND gate is shown below
Application of AND Gate
• The AND gate on this drill press
provides a safety feature which
prevent the operator from placing
a hand between the drill bit and
stock.
• To activate the drive, both
buttons must be pushed
simultaneously
“OR” Operation:
Describes events which can occur IF at LEAST
ONE of the other events is TRUE.
Since the Switches are in
parallel, the lamp will light
when A
OR B are closed, i.e. A or B =
“1” or Both “1”
A B L
OPEN OPEN OFF
OPEN CLOSE
D
ON
CLOSE
D
OPEN ON
CLOSE
D
CLOSE
D
ON
A B L
0 0 0
0 1 1
1 0 1
1 1 1
Application of OR Gates:
Here OR gate is used to detect an
exceed of temperature or pressure,
and then produce a command signal
for the system to take necessary
actions
“NOT” operation: Changes a statement from TRUE
to FALSE and vice–versa, i.e. inversion
The truth table for this operation
A L
OPEN ON
CLOSED OFF
A L
0 1
1 0
Application of NOT Gates:
A NOT GATE can be used to generate build a
square wave oscillator
Most times a NOT Gate gates ARE
not used on its own to perform a
function, rather its usually connected
to other gates to perform a function.
NOR GATE
A B C=A∪B
0 0 0
0 1 1
1 0 1
1 1 1
A B X
0 0 1
0 1 1
1 0 1
1 1 0
NAND GATE
XOR GATE A B X
0 0 0
0 1 1
1 0 1
1 1 0
XNOR GATE
A B X
0 0 1
0 1 0
1 0 0
1 1 1
BOOLEAN LOGIC IN PROGRAMMING
Boolean logic is fundamental to the design of computer hardware even if it isn’t the whole story.
The same holds true for programming.
In decision making, Boolean expression are applied while writing our programs
For example, a common task is making a decision using an IF statement
IF (A>0 AND A<10)
THEN do something
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