computer science 210 computer organization introduction to boolean algebra
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Computer Science 210Computer Organization
Introduction to
Boolean Algebra
George Boole
• English mathematician (1815-1864)
• Boolean algebra– Logic– Set Theory– Digital circuits– Programming: Conditions in while and if
statements
Boolean Constants
In Boolean algebra, there are only two constants, true and false
Boolean constant
Binary digit
State of a switch
Voltage level
true 1 On +5V
false 0 Off 0V
Boolean Variables
Boolean variables are variables that store values that are Boolean constants.
Let A be true
Let B be false
Etc.
Boolean Operator AND
If A and B are Boolean variables (or expressions) then
A AND B
is true if and only if both A and B are true.
Boolean Operator AND
If A and B are Boolean variables (or expressions) then
A AND B
is false if and only if either A or B are false or they’re both false.
Boolean Operator AND
We denote the AND operation like multiplication in ordinary algebra:
AB or A.B
Boolean Operator OR
If A and B are Boolean variables (or expressions) then
A OR B
is true if and only if at least one of A and B is true.
Boolean Operator OR
If A and B are Boolean variables (or expressions) then
A OR B
is false if and only if both A and B are false.
Boolean Operator OR
We denote the OR operation like addition in ordinary algebra:
A+B
Boolean Operator NOT
If A is a Boolean variable (or expression) then
NOT A
has the opposite value from A.
Boolean Operator NOT
We denote the NOT operation by putting a bar over the variable (or expression)
_A
Boolean Expressions
• As with ordinary algebra, a Boolean expression is a well-formed expression made from– Boolean constants
– Boolean variables
– Operators AND, OR and NOT
– Parentheses
• Example: __ ____ AB + (A+C)B
Evaluating a Boolean expression
At any time, the value of a BE can be computed using the current values of the variables.
__AB + (CD)
Let A = trueLet B = falseLet C = trueLet D = false
Then the resulting value is true
Operator precedence
NOT comes first, then AND, and finally OR
(Like arithmetic negation, product, and addition)
A + BC is not the same as
(A + B)C
Evaluating a Boolean expressionUnlike ordinary algebra, for a BE, there are only finitely many possible assignments of values to the variables; so, theoretically, we can make a table, called a truth table, that shows the value of the BE for every possible set of values of the variables.
For convenience, use 0 = false 1 = true
Truth Table for AND
A B AB
0 0 0
0 1 0
1 0 0
1 1 1
Truth Table for OR
A B A+B
0 0 0
0 1 1
1 0 1
1 1 1
Truth Table for NOT
A
_
A
0 1
1 0
Filling in a Truth Table
• If there are N variables, there are 2N possible combinations of values
• Thus, there are 2N rows in the truth table
• Fill in the values by counting up from 0 in binary
Construct a truth table for _ ___ E = AB + (A+C)B
Example
A B C
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
_ ___ E = AB + (A+C)B
Assign the values of the variables first
A B C
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
_
B
1
1
0
0
1
1
0
0
_ ___ E = AB + (A+C)B
Then add columns for each operation
A B C
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
_
B
1
1
0
0
1
1
0
0
_ ___ E = AB + (A+C)B _
AB
0
0
0
0
1
1
0
0
A B C
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
_
B
1
1
0
0
1
1
0
0
_ ___ E = AB + (A+C)B _
AB
0
0
0
0
1
1
0
0
A+C
0
1
0
1
1
1
1
1
A B C
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
_
B
1
1
0
0
1
1
0
0
_ ___ E = AB + (A+C)B _
AB
0
0
0
0
1
1
0
0
A+C
0
1
0
1
1
1
1
1
___
(A+C)
1
0
1
0
0
0
0
0
A B C
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
_
B
1
1
0
0
1
1
0
0
_ ___ E = AB + (A+C)B _
AB
0
0
0
0
1
1
0
0
A+C
0
1
0
1
1
1
1
1
___
(A+C)
1
0
1
0
0
0
0
0
___
(A+C)B
0
0
1
0
0
0
0
0
A B C
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
_
B
1
1
0
0
1
1
0
0
_ ___ E = AB + (A+C)B _
AB
0
0
0
0
1
1
0
0
A+C
0
1
0
1
1
1
1
1
___
(A+C)
1
0
1
0
0
0
0
0
___
(A+C)B
0
0
1
0
0
0
0
0
E
0
0
1
0
1
1
0
0
Laws of Boolean Algebra
Identity Zero ElementA+0=A, A.1=A A.0=0, A+1=1
Idempotent CommutativeA+A=A, AA=A A+B=B+A, AB=BA
Associative Distributive(A+B)+C=A+(B+C) A(B+C)=AB+AC(AB)C=A(BC) A+BC=(A+B)(A+C)
Laws of Boolean Algebra
Absorption DeMorganA+AB=A, A+B
____ = A
_ B_
,A(A+B)=A AB
___ = A
_ +B
_
Complement Double ComplementA+A
_ =1, AA
_ =0 A
__ = A
Boolean Expression Simplification
(A+B_
)C + AC_
+ (B+C____
)= (A+B
_ )C + AC
_ + B
_ C_
(DeMorgan)= (A+B
_ )C + (A+B
_ )C
_ (Distributive)
= (A+B_
)(C+C_
) (Distributive)= (A+B
_ )1 (Complement)
= A+B_
(Identity)
Boolean Expression Simplification
A_
B_
+ AB_
+ AB= A
_ B_
+ AB_
+ AB_
+ AB (Idempotent)= B
_ (A
_ +A) + A(B
_ +B) (Distributive)
= B_
1 + A 1 (Complement)= B
_ + A (Identity)