Professor : Dr. A. Karmouch, office CBY A508
Mid-term exam: Saturday March 1, 2014(10:00-11:30)
Chapter 2
Boolean Algebra and Logic Gates
2
Binary Logic
• Binary logic deals with
1 - Variables that can take on two discrete values
Values can be called True, False, yes, no, etc.
2 - Operations that assume LOGICAL Meaning
Binary logic is equivalent to Boolean algebra
Boolean Algebra
•Basic mathematics required for the description of digital circuits
• used to describe the different interconnections of digital circuits
• the variable used in the Boolean algebra are called
Boolean variables
We will study two-valued Boolean algebra and functions with simplifications using basic Boolean
Identities
Two-valued Boolean Algebra
• It consists of
1- Boolean Variables
- Designated by letters of the alphabet such as A, B, C, x, y, z etc.
- Each variable can have two and only two distinct
values: 1 and 0 (True, False)
- Can be a Function of some other Boolean variables
(F=ABC)
2- Boolean Operations
-There are three Basic logical operations:
AND, OR, and NOT
Basic Boolean Operations- AND operation
• Represented by a dot or by the absence of an operator
Example: read: x.y = or
xy=z
x AND y is equal to
z
Interpretation: Z = 1 if and only if
x= 1 AND y= 1
Otherwise z = 0
Truth table:
Don’t confuse this with binary multiplication operation x
y
xy
0
0
0
0
1
0
1
0
0
1
1
1
Truth table gives the value of z for all possible values of x and y
Basic Boolean Operations- OR operation
• Represented by a plus sign (+)
Example: read: x+y=z x OR y is equal to
z
Interpretation: z = 1 if x= 1 or if y= 1 or if both x =1 and y =1. z = 0 if x = 0 and y=1
Truth table:
Don’t confuse this with binary addition operation
x
y
x+y
0
0
0
0
1
1
1
0
1
1
1
1
Truth table gives the value of z for all possible values for x and y
Basic Boolean Operations- NOT operation
• Represented by a prime or an overbar (also called complement)
Example: read: x’ = z (or x = z)
Not x
is equal to
z
Interpretation: z = “what x is not” x= 1 then z=0; x= 0 then z=1
Truth table: x x’
0
1
1
0
Truth table gives the value of z for all possible values for x
Binary Logic and Binary Signals
• For simplicity, we often still write digits instead: – 1 is true
– 0 is false
• We will use this interpretation along with special operations to design functions and logic circuits for doing arbitrary computations.
Logic Gates
• Logic gates are electronic circuits that operate on one or more input signal to produce an output signal
•Basic operations can be implemented in hardware using a
Basic logic gate.
–Symbols for each of the logic gates are shown below.
–These gates output the product, sum or complement of their inputs Logic Operation:
Representation:
Logic gate:
AND (product) of two inputs
x.y, or xy
OR (sum) of two inputs x+y NOT
(complement)
With one input x’ Gates with Multiple Inputs
• AND and OR Gates may have more than
2 input signals
Binary Signals
•Computers use voltages to represent information.
•Two voltage levels are used to represent a binary value
“1” and “0”
• Some digital systems for example may define that:
- Binary ‘0” is equal to 0 Volt
- Binary “1” is equal to 4 Volt
It’s convenient for us to translate these
Volts
4
1
voltages into values 1 and 0.
0
0
Binary Logic and Binary Signals
•It’s also possible to think of voltages as representing two logical values, true and false.
These logical values are called Boolean values Volts
4
