Boolean Algebra & Minimization-1

Boolean Algebra

Boolean algebra is an algebraic structure defined on a set of elements together with two binary operators (+) and (.)

• A variable is a symbol, for example, Α, used to represent a logical quantity, whose value can be 0 or 1.
• The complement of a variable is the inverse of a variable and is represented by an overbar, for example ‘ .’
• A literal is a variable or the complement of a variable.

Closure: For any x and y in the alphabet A, x + y and x.y are also in A.

Boolean Value

The value of the Boolean variable can be either 1 or 0. The base for this number system is 2.

Boolean Operators

There are four Boolean operators

1. AND (∙) operator = The output will be like Y= (A.B)

A	B	Y
0	0	0
0	1	0
1	0	0
1	1	1

1. OR (+) operator= The output will be like Y= (A + B)

A	B	Y
0	0	0
0	1	1
1	0	1
1	1	1

1. NOT (A’) operator= The output will be like Y= Complement of A

A	Y
0	1
1	0

1. XOR (⊕) operator = The output will be like

A	B	Y
0	0	0
0	1	1
1	0	1
1	1	0

It is also known as inequality checking gate.

A	B	Y
0	0	1
1	0	0
0	1	0
1	1	1

It is also known as equality checking gate.

Formulas for GATE Computer Science Engineering – Digital Logic

Operator Precedence

The operator for evaluating Boolean expression is

1. Parenthesis
2. NOT
3. AND
4. OR.

Duality

If an expression contains only the operations AND, OR and NOT. Then, the dual of that expression is obtained by replacing each AND by OR, each OR by AND, all occurrences of 1 by 0, and all occurrences of 0 by 1. Principle of duality is useful in determining the complement of a function.

Example: Logic expression: (x •y’ •z) + (x •y •z’ ) + (y •z) + 0 ,

Duality of above logic expression is: (x + y’ + z) • (x + y + z’ ) • (y + z) • 1

Literals remain same while calculating the above expression.

Boolean Function

• Any Boolean functions can be formed from binary variables and the Boolean operators •, +, and ‘ (for AND, OR, and NOT, respectively).
• For a given value of variable, the function can take only one value either 0 or 1.
• A Boolean function can be shown by a truth table. To show a function in a truth table we need a list of the 2ⁿ combinations of 1’s and 0’s of the n binary variables and a column showing the combinations for which the function is equal to 1 or 0. So, the table will have 2ⁿ rows and columns for each input variable and tile final output.
• A function can be specified or represented in any of the following ways:
  • A truth table
  • A circuit
  • A Boolean expression
  • SOP (Sum Of Products)
  • POS (Product of Sums)
  • Canonical SOP
  • Canonical POS
• Important Boolean operations over Boolean values:

Table of Some Basic Theorems

Important Theorems used in Simplification

• NOT-Operation theorem:
• AND-Operation theorem:
• OR-Operation theorem:

• Distribution theorem: A + BC = A (A + B)(A + C)

Note:

• Demorgan’s Theorem:
• Transposition Theorem: (A + B) (A + C) = A + BC
• Consensus Theorem: This theorem is used to eliminate redundant term. It is applicable only when if a boolean function contains three variables. Each variable used two times. Only one variable is complemented or uncomplemented. Then the related terms so that complemented or uncomplemented variable is the answer.

Universal Logic Gate: A gate is considered as universal logic gate if it is capable to obtain all the operations just by using that single gate. The two universal logic agtes are NAND, NOR.

NAND: AND followed by NOT

it is represented by (↑).

The output will be like Y=

The truth table will be

A	B	Y
0	0	1
1	0	1
0	1	1
1	1	0

NOR:OR followed BY NOT

It is represented by (↓).

The truth table will be

A	B	Y
0	0	1
0	1	0
1	0	0
1	1	0

No of Gate Required:

Operations	NAND Gates Needed	NOR Gates Needed
NOT	1	1
OR	3	2
AND	2	3
XOR	4	5
ExNOR	5	4

Thanks