Chapter 2 Boolean Algebra and Minimization Techniques

2 Chapter Objectives Understand the concept of Boolean algebra Understand the concept of Boolean algebra Understand the logic operations available in Boolean algebra Understand the logic operations available in Boolean algebra Understand the basic laws of Boolean algebra Understand the basic laws of Boolean algebra Understand the DeMorgan’s theorems and Karnaugh map technique Understand the DeMorgan’s theorems and Karnaugh map technique

3 Boolean Algebra Mathematician George Boole defined Boolean algebra as the technique, which provides the rules for carrying out the logical operations. Mathematician George Boole defined Boolean algebra as the technique, which provides the rules for carrying out the logical operations. The logical operations available in Boolean algebra are: The logical operations available in Boolean algebra are: Logical AND operation: It is given as Y = A.B for two Boolean variables A and B. The values of AND operation for different values of A and B are: Logical AND operation: It is given as Y = A.B for two Boolean variables A and B. The values of AND operation for different values of A and B are: If A=0,B=0 then A.B = 0 If A=0,B=0 then A.B = 0 If A=1,B=0 then A.B = 0 If A=1,B=0 then A.B = 0 If A=0,B=1 then A.B = 0 If A=0,B=1 then A.B = 0 If A=1,B=1 then A.B = 1 If A=1,B=1 then A.B = 1

4 Boolean Algebra(contd..) Logical OR operation: It is given as Y=A+B. The values of OR operation for different values of A and B are: Logical OR operation: It is given as Y=A+B. The values of OR operation for different values of A and B are: If A=0, B=0 then A+B = 0 If A=0, B=0 then A+B = 0 If A=1, B=0 then A+B = 1 If A=1, B=0 then A+B = 1 If A=0, B=1 then A+B = 1 If A=0, B=1 then A+B = 1 If A=1, B=1 then A+B = 1 If A=1, B=1 then A+B = 1 Logical complementation operation: It is also known as the NOT operation and it converts the logical 1 to 0 and logical 0 to 1. Logical complementation operation: It is also known as the NOT operation and it converts the logical 1 to 0 and logical 0 to 1. If A is a Boolean variable, then the complement of A is represented by A* or A′. If A is a Boolean variable, then the complement of A is represented by A* or A′. Alternatively, a bar over the variable is used to represent the complement of the variable. Alternatively, a bar over the variable is used to represent the complement of the variable.

5 Basic Laws of Boolean Algebra The basic laws of Boolean algebra are used to provide mathematical expressions for logical operations. The basic laws of Boolean algebra are used to provide mathematical expressions for logical operations. The Boolean rules define only two variables, binary 1 and binary 0. The basic rules of Boolean algebra are: The Boolean rules define only two variables, binary 1 and binary 0. The basic rules of Boolean algebra are: Boolean addition Boolean addition = = = = = = = = 1 Boolean multiplication Boolean multiplication 0.0 = = = = = = = = 1

6 Properties of Boolean Algebra The properties of Boolean algebra depend on the following operators: The properties of Boolean algebra depend on the following operators: Two binary operators denoted by the symbols ‘+’ and ‘.’ Two binary operators denoted by the symbols ‘+’ and ‘.’ One unary operator denoted by either the bar (¯) symbol or prime (') symbol One unary operator denoted by either the bar (¯) symbol or prime (') symbol The various properties of Boolean algebra are: The various properties of Boolean algebra are: Commutative property: The order of a Boolean operation conducted on the variables makes no difference on the result. For example: Commutative property: The order of a Boolean operation conducted on the variables makes no difference on the result. For example: Boolean addition is commutative, i.e. A+B = B+A Boolean addition is commutative, i.e. A+B = B+A Boolean multiplication is commutative, i.e. A.B = B.A Boolean multiplication is commutative, i.e. A.B = B.A

7 Properties of Boolean Algebra (Contd..) Associative property: A Boolean operation conducted on several variables provides the same result irrespective of their grouping. For example: Associative property: A Boolean operation conducted on several variables provides the same result irrespective of their grouping. For example: Boolean addition is associative. It is represented as: Boolean addition is associative. It is represented as: A+(B+C) = (A+B)+C Boolean multiplication is associative. It is represented as: Boolean multiplication is associative. It is represented as: A.(B.C) = (A.B).C Distributive Property: Boolean algebra is distributive for both addition and multiplication operations. Distributive Property: Boolean algebra is distributive for both addition and multiplication operations. Boolean addition is distributive over the Boolean multiplication. It is represented as: Boolean addition is distributive over the Boolean multiplication. It is represented as: A+B.C= (A+B).(A+C) Boolean multiplication is distributive over the Boolean addition. It is represented as: Boolean multiplication is distributive over the Boolean addition. It is represented as: A.(B+C)= A.B+A.C

8 DeMorgan’s Theorems According to DeMorgan’s first theorem, the complement of the product of two binary variables is equal to the sum of the complements of two binary variables. According to DeMorgan’s first theorem, the complement of the product of two binary variables is equal to the sum of the complements of two binary variables. It can be stated as: It can be stated as: (AB) * = A * + B * According to DeMorgan’s second theorem, the complement of the sum of the two binary variables is equal to the product of the complements of two binary variables. According to DeMorgan’s second theorem, the complement of the sum of the two binary variables is equal to the product of the complements of two binary variables. It can be stated as: It can be stated as: (A + B) * = A *. B *

9 Sum of Products and Product of Sums Sum of Products (SOP) and Product of Sums (POS) are the logical variables that are used to express logical functions. Sum of Products (SOP) and Product of Sums (POS) are the logical variables that are used to express logical functions. SOP: It is the logical sum of two or more logical product term. Basically, it is an OR operation of AND operated variables. For example: SOP: It is the logical sum of two or more logical product term. Basically, it is an OR operation of AND operated variables. For example: Y = AB+BC+AC Y = AB+BC+AC POS: It is the logical product of two or more logical sum terms. Basically, it is an AND operation of OR operated variables. For example: POS: It is the logical product of two or more logical sum terms. Basically, it is an AND operation of OR operated variables. For example: Y = (A+B)(B+C)(A+C) Y = (A+B)(B+C)(A+C)

10 Karnaugh Map Karnaugh map is a systematic method for simplifying and manipulating switching expressions. Karnaugh map is a systematic method for simplifying and manipulating switching expressions. It is used to represent the information contained in a truth table or the information available in the form of POS or SOP. The characteristics of a K-map are: It is used to represent the information contained in a truth table or the information available in the form of POS or SOP. The characteristics of a K-map are: In an n-variable K-map, there are 2 n cells and each cell matched to one combination of n variables. In an n-variable K-map, there are 2 n cells and each cell matched to one combination of n variables. In an n-variable K-map, a collection of 2 m cells and each adjacent to m cells is known as a group.This group can be expressed by a product containing n-m variables where n is the number of variables in the K-map. In an n-variable K-map, a collection of 2 m cells and each adjacent to m cells is known as a group.This group can be expressed by a product containing n-m variables where n is the number of variables in the K-map.