Boolean Algebra & Logic Gates

By : Ali Mustafa

Digital Logic Gates

• There are three fundamental logical operations, from which all other functions, no matter how complex, can be derived. These Basic functions are named:

–AND

–OR

–NOT (INVERTER)

AND Gate

• Represented by any of the following notations:

– X AND Y

– X . Y

– X Y

• Function definition:

Z = 1 only if X=Y=1

0 otherwise

OR Gate • Represented by any of the following

notations:

– X OR Y

– X + Y

• Function definition:

Z = 1 if X=1 or Y =1 or both X=Y=1

0 if X=Y=0

NOT (Inverter) Gate

• Represented by a bar over the variable

• Function definition:

It is also called complement operation , as it changes 1’s to 0’s and 0’s to 1’s.

Logic gates timing Diagram• Timing diagrams illustrate the response of any gate to all

possible input signal combinations.

• The horizontal axis of the timing diagram represents time and the vertical axis represents the signal as it changes between the two possible voltage levels 1 or 0

Other Gates

Others Gates

Digital Logic

• How to describe Digital logic system?

We have two methods: • Truth Table • Boolean Expression

TRUTH TABLE • A Truth Table is a table of combinations of the binary variables

showing the relationship between the different values that the input variables take and the result of the operation (output).

• The number of rows in the Truth Table is where n = number of input variables in the function.

• The binary combinations are obtained from the binary number by counting from 0 to

• Example: AND gate with 2 inputs

– n=2

– The truth table has 2 rows = 4

– The binary combinations is from

0 to (22-1=(3)) [00,01,10,11]

BOOLEAN EXPRESSIONS

• We can use these basic operations to form more complex expressions:

• Some terminology and notation:– f is the name of the function.

– (x,y,z) are the input variables, each representing 1 or 0. Listing the inputs is optional, but sometimes helpful.

– A literalis any occurrence of an input variable or its complement. The function above has four literals: x, y’, z, and x’.

How to get the Boolean Expression from the truth table?

Boolean Expressions From Truth Tables

• Each 1 in the output of a truth table specifies one term in the corresponding booleanexpression.

Example

• Find boolean expression?

Example Solution

Basic Logic gates

• We have defined three basic logic gates and operators

• Also, we could build any digital circuit from those basic logic gates.

• In digital Logic, we are not using normal mathematics we are using Boolean algebra

So, we need to know the laws & rules of Boolean Algebra

Boolean Algebra

• What’s the difference between the Boolean Algebra and arithmetic algebra?

The First obvious difference that in Boolean algebra we have only (+) and (.) operators we don’t have subtraction(-) or division(/) like math.

Binary Logic

• You should distinguish between binary logic and binary arithmetic.

– Arithmetic variables are numbers that consist of many digits.

Arithmetic

• A binary logic variable is always either 1 or 0.

Binary

Laws & Rules of Boolean Algebra

• The basic laws of Boolean algebra:

–The commutative law

–The associative law

–The distributive law

Commutative Law

• The commutative law of addition for two variables is written as: A+B = B+A

• The commutative law of multiplication for two variables is written as: AB = BA

Associative Law

• The associative law of addition for 3 variables is written as: A+(B+C) = (A+B)+C

• The associative law of multiplication for 3 variables is written as: A(BC) = (AB)C

Distributive Law • The distributive law for multiplication as

follows: A(B+C) = AB + AC

• The distributive law for addition is as follows A+(B.C) = (A+B)(A+C)

Basic Theorems of Boolean Algebra

Basic Theorems of Boolean Algebra

OR Laws AND Laws

Inversion Law

Theorem 1(a)

Proof X + X = XSolution

X + X = (X + X ) . 1 X.1=1

=(X + X)(X + X’) X + X’=1

=X + XX’ Dist X + YZ= (X+Y)(X+Z)

=X + 0 X.X’=0

=X

Theorem 1(b)

Prove X.X = XSolution:

X.X = XX + 0 X + 0 =X

=XX + XX’ XX’= 0

=X(X + X’)

=X(1) X + X’ =1

=X

Self Tasks (Theorems)

– X + 1 = 1

– X.0 = 0

– X + XY= X

Duality Principle • A Boolean equation remains valid if we take

the dual of the expressions on both sides of the equals sign

• Dual of expression it means,

– Interchange 1’s with 0’s (and Vice-versa)

– Interchange AND (.) with OR (+) (and Vice-versa)

DeMorgan’s Law Theorem 1: NAND = Bubbled OR Complement of product is equal to addition of the compliments.

Theorem 2: NOR = Bubbled AND Complement of sum is equal to product of the compliments.

Truth Tables for DeMorgan’s

Example• Get the logic function from the following truth table

and implement it using basic logic gates (AND, OR, NOT)

Simplification of the logic function

• A´B´ + A´B + AB´

Solution:

A´(B´ + B) + AB´

A’(1) + AB´

(A’ + A)(A’ + B’) Hint A+AB Dist

1 (A’ + B’)

(AB)’ DeMorgan Law(1 NAND Gate only)

From 7 gates using simplification rule can be optimized to 1 gate

Home Task: Simplify the following Expressions

• Find the dual of the following expressions

1. A + AB = A

2. A + A’B = A + B

3. A + A’ = 1

4. (A + B)(A + C) = A + BC

Home Task: Simplify the following Expressions

• Prove the following binary expressions ( Using

Postulates)

1. A + AB = A

2. (A + B)(A + C) = A + BC

3. (A + B’ + AB)(A + B’)(A’B) = 0

4. AB + ABC + AB’ = A

5. AB’ + A’B + A’B’

Fewer Gates

XZYX F

Algebraic Manipulation

XZZYX YZ X F