Boolean Algebra
Ver. 1.4
© 2010 - Claudio Fornaro
Boolean Algebra
George Boole was an English mathematician of XIX century
Boolean Algebra can operate on logic (or Boolean) variables that can assume just 2 values:
1/0, true/false, on/off, closed/open
Usually value 1 is associated to True, while 0 to False
3
Boolean Algebra
Boolean algebra is suited for representing
“ binary events ”, that is conditions that can assume just 2 values
Example
A light bulb can be turned on or off
Functions operating on Boolean variables are called Boolean functions and can only produce values 0 or 1
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Boolean Algebra
A Boolean function F, computed on the Boolean variables v
1,v
2,...,v
nis denoted as:
A Boolean function can be defined in many ways, among them the truth table:
a list of the values that F produces for every possible combination of the variables
) , ,
,
( v
1v
2v
nF
Boolean Algebra
Example
F(v
1,v
2,v
3) can be defined as shown on the right side:
Each Boolean variable can assume only 2 values, thus n variables can produce 2
npossible combinations, and each one is a row of the table
v3 v2 v1 F 0
0 0 0
0 1
1 1 0
0 1 1
0 1
0 0 1
1 0 0
0 1
0 1 1
1 1 1
0 1
1 1
Boolean Algebra
Description of an event by using a Boolean function
To pass a certain exam, a student must match one of the following conditions:
s/he passes the written pre-test and then the oral discussion
s/he passes the written test and then the oral discussion
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Boolean Algebra
Each event is either independent or
dependent (from the independent events)
Each event is assigned a Boolean variable:
a pre-test, 1=passed b written test, 1=passed c oral test, 1=passed e exam, 1=passed
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Boolean Algebra
3 variables allow 8 (2
3) combinations
The truth table for the Boolean function “passing the exam” then is:
a b c e 0
0 0 0
0 1
0 0 0
0 1 1
0 1
0 1 1
1 0 0
0 1
0 1 1
1 1 1
0 1
0 1
Meaning of this row:
the student passed pre-test (a=1), didn’t pass (or try) written test (b=0) and passed oral test (c=1), thus s/he passes the whole exam (e=1)
Boolean Algebra
After looking carefully, condition a = 0, b = 0, and c = 1 cannot happen, because the oral examination can be tried only when the pre- test or the written test has been passed (or both)
For that combination the exam is not passed, but this is an impossible condition, so it could be useful to indicate a don’t care condition (usually represented by a “–”). A don’t care mark will be eventually replaced by either a 0 or a 1 as needed in a next stage
Logic Operators
Boolean variables can be combined by using Boolean (or logic) operators into Boolean (or logic) expressions
The result of a logic operation is still a logic value (true or false)
The most important operators are:
AND, OR, NOT
EXOR
NAND, NOR, EXNOR
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Logic Operators
Operator AND (logic product)
symbol: • (like the algebraic product), it is often omitted
it combines 2 values and produces a result following the rules below:
0 0 = 0 0 1 = 0 1 0 = 0 1 1 = 1
Result is True if both operands are True
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Logic Operators
Example
“I go to sea if it is sunny AND it is warm”
X=“it is sunny”
Y=“it is warm”
Z=“I go to sea”
Z = X • Y
Both condition must be true to have that “I go to the sea” be true
Logic Operators
Operator OR (Inclusive OR, logic sum)
symbol: + (like the algebraic sum)
it combines 2 values and produces a result following the rules below:
0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1
Result is True if at least one operand is True
“I take car if it is raining OR it is cold”
Logic Operators
Example
“I use car if it is raining OR it is cold”
X=“it is raining”
Y=“it is cold”
Z=“I use car”
Z = X + Y
At least one condition must be true to have that “I use car” be true
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Logic Operators
Operator NOT
symbol: a little line over the variable or the expression to be negated, or just a ’ sign
it applies to just one value and produces a result following the rules below:
0 = 1 1 = 0
Result is True if the operand is False, and vice versa
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Logic Operators
Operator EX-OR (Exclusive OR)
symbol:
it combines 2 values and produces a result following the rules below:
0 0 = 0 0 1 = 1 1 0 = 1 1 1 = 0
Result is True if just one operand is True
Logic Expressions
A logic (or Boolean) expression is composed by:
logic variables
constants 0 and 1
logic operators
parentheses Examples
a b c
e c ) e a c(d b
a
Logic Expressions
In a logic expression the operator with the maximum priority is NOT, followed by AND and then OR
This means that the following expressions are equivalent
F = a + b c F = a + (b c)
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Logic Expressions
Two expressions F
1and F
2are said equivalent when each input variable combination produce the same output value on both functions (they have the same truth tables)
Examples
2 1
1
x F
x F
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Logic Expressions
Two expressions F
1and F
2are said
complementary when each input variable combination produce an opposite output value on both functions (they have
complementary truth tables)
Examples
b a F
b a F
2 1
Logic Expressions
Two expressions F
1and F
2are said dual when:
each OR of F1 corresponds to an AND of F2 and vice versa
each 1 of F1corresponds to an 0 of F2and v.v.
the order of operator evaluation is the same
Example
)) 0 ( (
) 1 (
2 1
c b a F
c b a F
note the added parentheses
Logic expression reduction
A reduced expression is faster to be
evaluated and the corresponding circuit is smaller, faster, and less power consuming
Reduction can be accomplished in many ways, among them:
Boolean algebra axioms and theorems
Karnaugh maps
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Boolean Algebra Axioms and Th.
Theorem of duality:
if an equivalence holds, the dual equivalence holds as well
Axioms and theorems
1. x 0 = 0 x + 1 = 1
2. x 1 = x x + 0 = x
3. x x = 0 x + x = 1
4. x x = x x + x = x
5. x y = y x x + y = y + x
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Boolean Algebra Axioms and Th.
6. xyz = (xy) z = x+y+z = (x+y) + z =
= x (yz) = = x + (y+z) =
= y (xz) = y + (x+z)
7. De Morgan’s Theorem
xyz… = x+y+z+… x+y+z+… = xyz…
8. xy+xz = x(y+z) (x+y)(x+z) = x+yz
9. x+xy = x x(x+y) = x
10. xy + xy = x (x+y)(x+y) = x
11. x+xy = x+y x(x+y) = xy
12. xy = xy+xy = xy xy=xy=xy=xy+xy
Boolean Algebra Axioms and Th.
Some demonstrations
9. x+xy = x1+xy = x(1+y) = x1 = x
8. (x+y)(x+z) = xx+xz+xy+yz =
= x+xz+xy+yz =
= x1 +xz+xy+yz =
= x(1+z+y)+yz = x1+yz = x+yz
10. xy+xy = x(y+y) = x1 = x
11. x+xy = x1+xy = x(1+y)+xy = x+xy+xy =
= x+y rule 10
b.
Boolean Algebra Axioms and Th.
In the previous equalities, variables x , y, and z can be considered both as single variables or as full logic expressions E.g. from rule 1bis: (ab+cd) + 1 = 1 where the parentheses content is considered as x
Note that the complementary of xy is NOT xy, but is xy and this for the De Morgan’s Theorem is equal to x+y
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Reduce the following expressions
1. rule 11
rule 3b
rule 1b
2. rule 11b
rule 4 rule 1
Examples of reduction
1 1
) (
1
) (
) (
everything B A B A A
B A B A B B A
B A B A B A B A
C C B
C A BA
C B A B A
0
) (
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Another demonstration of rule 11
x+xy = x1+xy = x(y+y)+xy =
= xy+xy+xy = xy+xy+xy+xy = [rule 4b]
= x(y+y)+y(x+x) = x+y
Any term in OR may be duplicated, triplicated, etc.
following rule 4b (backward): x=x+x+x+…
Any term in AND may be duplicated, triplicated, etc. following rule 4a (backward): x=xxx…
Examples of reduction
Logic Functions and Truth Tables
To obtain the truth table from a logic function, the output value for each
combination of the input values must be computed
Example
Compute the truth table of the following function:
F(a,b,c) = (ab + b) c
Logic Functions and Truth Tables
abc ab b ab + b c (ab+b) c
000 0 1 1 1 1
001 0 1 1 0 0
010 0 0 0 1 0
011 0 0 0 0 0
100 0 1 1 1 1
101 0 1 1 0 0
110 1 0 1 1 1
111 1 0 1 0 0
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Expressions in Canonical Forms
SP Canonical Form
Logical sum of products (SP)
Each term (called minterm) is a product of all the variables (some could be affirmed, other negated)
Example
this expression is composed by 4 minterms F(a,b,c) = abc+abc+abc+abc
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Expressions in Canonical Forms
PS Canonical Form
Logical product of sums (PS)
Each term (called maxterm) is a sum of all the variables (some could be affirmed, other
negated)
Example
this expression is composed by 3 maxterms
F(a,b,c) = (a+b+c) (a+b+c) (a+b+c)
From Truth Table to SP
Because the function must give 1 for some specific combination of input values, for each row with output value 1:
write a product of all the variables (minterm)
Each minterm must give 1 for a specific combination of the input variables, so:
negate each variable whose value is 0 on the input combination corresponding to the minterm
Sum the minterms
From Truth Table to SP
Example
F(a,b,c) = abc+abc+abc+abc
a b c F
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 0
abc
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From Truth Table to PS
Because the function must give 0 for some specific combination of input values, for each row with output value 0:
write a sum of all the variables (maxterm)
Each maxterm must give 0 for a specific combination of the input variables, so:
negate each variable whose value is 1 on the input combination corresponding to the maxterm
Multiply the maxterms
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From Truth Table to PS
Example
F(a,b,c)=(a+b+c)(a+b+c)(a+b+c)(a+b+c)
a b c F
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 0
( a b c )
Expressions in Canonical Forms
To canonicalize an SP expression
If in a product a variable is missing (e.g. x) multiply the product by (x+x) and solve removing duplicates
Example
F(x,y,z) = xyz+yz+z
= xyz+(x+x)yz+(x+x)(y+y)z
= xyz+xyz+xyz+xyz+xyz+xyz+xyz
Expressions in Canonical Forms
To canonicalize a PS expression
If in a sum a variable is missing (e.g. x) add to the sum the term xx and solve removing duplicates
Example
F(x,y,z) = (x+y+z) (x+z) x
= (x+y+z) (x+yy+z) (x+yy+zz)
= (x+y+z) (x+y+z) (x+y+z) etc.
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Logic Circuits
A logical function can be implemented in a digital electronic circuit
Input variables are input signals
Output variables are output signals
Signals have just 2 values: low and high, associated to logic values 0 and 1
Logic operators corresponds to logic gates (e.g. in an AND gate, the output signal is high only when the 2 input signals are high)
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Logic Gates
AND OR
NOT
NAND =
NOR =
EXOR
EX-NOR =
Logic Gates
Multiple-input gates exists and are equivalent to more gates connected (thanks to the associative property), but are faster
A multiple-input EXOR gate gives 1 when the number of input with value 1 is odd
Logic Circuits
Example of logic circuit equivalent to a logic expression
F = a b + c
Note how operator priority is provided by the relative position of the gates
a b
c F
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Logic Circuits
Example a
F b c F ( a bc ) ( ab c a b )
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Examples
1.
Four doors (A, B, C, and D) separate two rooms. The doors are operated by 3
switches (X, Y, and Z) that, when pressed, close some doors: X closes A and C, Y closes B and D, Z closes B and C.
Draw the corresponding truth table that produces 1 only when all the doors are closed. Determine the corresponding
function with the least number of variables.
Examples
XYZ ABCD D 000 0000 0 001 0110 0 010 0101 0 011 0111 0 100 1010 0 101 1110 0 110 1111 1 111 1111 1
1=Door closed
(Note the truth table is the bold part, columns ABCD are just intermediate values)
D=XYZ+XYZ=XY
Examples
2.
Design a digital circuit with inputs:
1. a 2C value D composed by bits a and b
2. a selector: one bit s
and outputs:
1. a 2C value U composed by bits x and y
2. an overflow indicator: one bit w
that works in this way:
U
=
Dif
s=0
U=
–Dif
s=1
w
= 1 if the 2C operation produces an incorrect result
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