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Chapter#2

BINARY LOGIC AND BOOLEAN ALGEBRA

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INTRODUCTION

Logic gates are the basic building blocks of any digital circuit.
Logic Circuits are made up of various logic gates.
Logic Levels: There are two logic levels-0 & 1.

Logic Systems: Positive logic system & Negative Logic System.

Device	Logic 0	Logic 1
Switch	Off	On
Lamp	Off	On
Door	Open	Close
Level	Low	High

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Types of Logic Gates

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NOT GATE

A NOT gate accepts one input signal (0 or 1) and returns the opposite signal as output.
It is also known as inverter.

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AND GATE

The output of AND gate is high (‘1’) if all of its inputs are high (‘1’).
The output of AND gate is low (‘0’) if any one of its inputs is low (‘0’).

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OR GATE

The output of OR gate is high (‘1’) if any one of its inputs is high (‘1’).
The output of OR gate is low (‘0’) if all of its inputs are low (‘0’).

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NAND GATE

The NAND gate accepts two input signals.

If both are 1, the output is 0; otherwise, the output is 1

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NOR GATE

The NOR gate accepts two input signals

If both are 0, the output is 1; otherwise, the output is 0

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EX-OR or XOR GATE

An XOR gate accepts two input signals

If both are the same, the output is 0; otherwise, the output is 1.

Note the difference between the XOR gate and the OR gate; they differ only in one input situation
When both input signals are 1, the OR gate produces a 1 and the XOR produces a 0.
XOR is called the exclusive OR

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EX-NOR or X-NOR GATE

It is a combination of XOR gate followed by and inverter.
Its output is “1” if inputs are same and “0” if inputs are different.

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SUMMARY OF LOGIC GATES

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BUBBLED GATE

When input to a gate is given through inverter, the symbol can be made simpler by drawing a small circle at the input.
When a bubble is shown at the input, it is understood that the input to the gate goes through inverter.

Input through inverter Bubbled OR gate

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NAND AS UNIVERSAL GATE

All other gates can be realized using NAND gates.

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NAND AS UNIVERSAL GATE

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NAND AS UNIVERSAL GATE

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NAND AS UNIVERSAL GATE

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NOR AS UNIVERSAL GATE

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NOR AS UNIVERSAL GATE

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NOR AS UNIVERSAL GATE

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NOR AS UNIVERSAL GATE

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APPLICATIONS OF LOGIC GATES

Digital clock
Digital instruments
In process control equipments
Microprocessor
Microcomputer
Calculator
Cell Phone
Commercial equipments
Integrated Circuits

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BOOLEAN ALGEBRA

English Mathematician George Boole invented symbolic logic in 1854, which is known as Boolean Algebra.
Logic problems were solved with the help of this algebra.
BOOLEAN LOGIC OPERATIONS

Logical AND
Logical OR
Logical NOT

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LAWS OF BOOLEAN ALGEBRA

(a) A+0=A (b) A.1=A
(a) A+1=1 (b) A.0=0
(a) A+A=A (b) A.A=A
(a) A+A′=1 (b) A.A′=0
A′′=A

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PROPERTIES OF BOOLEAN ALGEBRA

COMMUTATIVE PROPERTY: If the sequence of variable is changed, there is no change in result.

(1) A+B=B+A OR is commutative

(2) A.B=B.A AND is commutative

2. ASSOCIATIVE PROPERTY: If variables are grouped in different ways there is no change in the result.

(1) A+(B+C)=(A+B)+C OR is associative

(2) A.(B.C)=(A.B).C AND is associative

3. DISTRIBUTIVE PROPERTY:

(I) A+BC=(A+B)(A+C) (2) A.(B+C)=AB+AC

Proof: LHS= A+BC

= A.1+BC

= A(1+B)+BC (⸪ 1+B=1)

= A.1+AB+BC

= A.(1+C)+AB+BC (⸪ 1+C=1)

= A.1+AC+AB+BC

= A.A+AC+AB+BC (⸪ A.A=A)

= A(A+C)+B(A+C)

= (A+B)(A+C)

=RHS

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PROPERTIES OF BOOLEAN ALGEBRA

4. ABSORPTION PROPERTY:

(I) A+AB=A

Proof: LHS= A+AB

= A(1+B)

= A(1)

= A

=RHS

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

Proof: LHS= A.(A+B)

= A.A+AB

= A+AB

= A(1+B)

= A.1

=A

= RHS

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PROPERTIES OF BOOLEAN ALGEBRA

(3) A+A′B=A+B

Proof: LHS= A+A′B

= (A+A′)(A+B) (⸪ A+BC=(A+B)(A+C))

= 1(A+B) (⸪ A+A′=1)

= (A+B)

=RHS

(4) A(A′+B)=AB

Proof: LHS= A(A′+B)

= A. A′+AB

= 0+AB (⸪ A. A′=0)

=AB

=RHS

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PROPERTIES OF BOOLEAN ALGEBRA

5. CONSENSUS PROPERTY:

(1) AB+A′C+BC=AB+ A′C

(2) (A+B)(A′+C)(B+C)=(A+B)(A′+C)

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EXAMPLES OF BOOLEAN ALGEBRA

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EXAMPLES OF BOOLEAN ALGEBRA

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EXAMPLES OF BOOLEAN ALGEBRA

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DE MORGAN’S THEOREMS

There are two theorems
THEOREM 1:

Complement of sum is equal to the product of the complements i.e

Left hand side is the complement of the sum. On the right hand side there is product instead the sum and each term is the complement of the variable.

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DE MORGAN’S THEOREMS

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DE MORGAN’S THEOREMS

THEOREM 2:

Complement of the product is the sum of the complements i.e

Left hand side is the complement of the product of the variables A & B. On the right hand side there is sum of the complements of variables A and B.

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DE MORGAN’S THEOREMS

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DE MORGAN’S THEOREMS

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DRAWING LOGIC CIRCUIT FROM BOOLEAN EXPRESSION (Q=A.B+B.C(B+C))

🡪 Our first step in simplification must be to write a Boolean expression for this circuit.

🡪This task is easily performed step by step if we start by writing sub-expressions at the output of each gate, corresponding to the respective input signals for each gate.

🡪 Remember that OR gates are equivalent to Boolean addition, while AND gates are equivalent to Boolean multiplication.

For example, I’ll write sub-expressions at the outputs of the first three gates:

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DRAWING LOGIC CIRCUIT FROM BOOLEAN EXPRESSION (Q=A.B+B.C(B+C))

then another sub-expression for the next gate:

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DRAWING LOGIC CIRCUIT FROM BOOLEAN EXPRESSION

Finally, the output (“Q”) is seen to be equal to the expression AB + BC(B + C):

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DRAWING LOGIC CIRCUIT FROM BOOLEAN EXPRESSION (REDUCING EXPRESSION)

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DRAWING LOGIC CIRCUIT FROM BOOLEAN EXPRESSION (REDUCING EXPRESSION)

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DRAWING LOGIC CIRCUIT FROM BOOLEAN EXPRESSION (REDUCING EXPRESSION)

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DRAWING LOGIC CIRCUIT FROM BOOLEAN EXPRESSION (MORE EXAMPLES)

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DRAWING LOGIC CIRCUIT FROM BOOLEAN EXPRESSION (MORE EXAMPLES)