![Distributive law Proof
Proof:
A (B + C) = A (B*1) + A (C*1) → since 1 * B = B, 1 * C = C
= [(AB)*(A*1)] + [(AC) *(A*1)]
=[(AB) * A] + [(AC) *A]
= (A +1) (AB + AC)
= (AB +AC) → since 1 + A = 1
Hence, distributive law is
proved.](https://image.slidesharecdn.com/unit2digitalfundamentalsbooleanfunc-240829185256-22ec1185/85/Unit-2-digital-fundamentals-boolean-func-pptx-13-320.jpg)
Unit 2 digital fundamentals boolean func.pptx
- 1. Boolean Functions Mr V Nivedan Assistant Professor Department of Electronics Sri Ramakrishna College of Arts and
- 2. Boolean Algebraic Laws The logic symbols “0” and “1” being used to represent a digital input or output, we can also use them as constants for a permanently “Open” or “Closed” circuit or contact respectively. A set of rules or Laws of Boolean Algebra expressions have been invented to help reduce the number of logic gates needed to perform a particular logic operation resulting in a list of functions or theorems known commonly as the Laws of Boolean Algebra.
- 3. Boolean Algebraic Laws Switching circuits are also called logic circuits, gates circuits and digital circuits. Switching algebra is also called Boolean algebra. Boolean algebra is a system of mathematical logic. It is an algebraic system consisting of the set of elements (0,1), two binary operators called OR and AND and unary operator called NOT. It is the basic mathematical tool in the analysis and synthesis of switching circuits. It is a way to express logic functions algebraically. Any complex logic can be expressed by a Boolean function. The Boolean algebra is governed by certain well developed rules and laws.
- 4. Boolean Algebraic Laws Boolean Algebra is use to analyse digital gates and circuits. These “Laws of Boolean” i s u s e d to both reduce and simplify a complex Boolean expression in an attempt to reduce the number of logic gates required. The main aim of any logic design is to simplify the logic as much as possible so that the final implementation will become easy.
- 5. Boolean Algebraic Laws In order to simplify the logic, the Boolean equations and expressions representing that logic must be simplified. So, to simplify the Boolean equations and expression, there are some laws and theorems proposed. Using these laws and theorems, it becomes very easy to simplify or reduce the logical complexities of any Boolean expression or function.
- 7. Boolean Algebraic Laws Basic Laws and Proofs: The basic rules and laws of Boolean algebraic system are known as “Laws of Boolean algebra”. Some of the basic laws (rules) of the Boolean algebra are i. Associative law iv. Absorption law ii. Distributive law v. Consensus law iii. Commutative law
- 8. Associative Law Associate Law of Addition Statement: Associative law of addition states that OR ing more than two variables i.e. mathematical addition operation performed on variables will return the same value irrespective of the grouping of variables in an equation.It involve in swapping of variables in groups. The Associative law using OR operator can be written as A+(B+C) = (A+B)+C
- 9. Associate Law Associate Law of Multiplication Statement: Associative law of multiplication states that ANDing more than two variables i.e. mathematical multiplication operation performed on variables will return the same value irrespective of the grouping of variables in an equation. The Associative law using AND operator can be written as A * (B * C) = (A * B) * C
- 10. Distributive law Statement 1 : The multiplication of two variables and adding the result with a variable will result in same value as multiplication of addition of the variable with individual variables. In other words, ANDing two variables and ORing the result with another variable is equal to AND of ORing of the variable with the two individual variables. Distributive law can be written as A + BC = (A + B)(A + C) This is called OR distributes over AND.
- 11. Distributive law Proof Proof: If A, B and C are three variables then A + BC = A * 1 + BC → since A*1 = A = A (1 + B)+ BC → since 1 + B = 1 = A * 1 + AB + BC = A *(1 + C) + AB + BC → since A*A = A*1 = A = A *(A + C) + B (A + C) = (A + C) (A + B) A + BC = (A + B) (A + C) Hence, distributive law is proved.
- 12. Distributive law Statement 2: The addition of two variables and multiplying the result with a variable will result in same value as addition of multiplication of the variable with individual variables. In other words, ORing two variables and ANDing the result with another variable is equal to OR of ANDing of the variable with the two individual variables. Distributive law can be written as A (B+C) = (A B) + (A C)
- 13. Distributive law Proof Proof: A (B + C) = A (B*1) + A (C*1) → since 1 * B = B, 1 * C = C = [(AB)*(A*1)] + [(AC) *(A*1)] =[(AB) * A] + [(AC) *A] = (A +1) (AB + AC) = (AB +AC) → since 1 + A = 1 Hence, distributive law is proved.
- 14. Example for distributive law Take three variables 0, 1 and 0, then According to distributive law, 0 (1 + 0) = (0*1) + (0*0) 0 (1) = (0) + (0) 0 = 0
- 15. Commutative law Statement: Commutative law states that the inter-changing of the order of operands in a Boolean equation does not change its result. A + B = B + A ● Using OR operator → ● Using AND operator → A * B = B * A This law is also has more priority in Boolean algebra. Example: Take 2 variables 1 and 0, then 1 + 0 = 0 + 1 1 = 1 Similarly, 1 * 0 = 0 * 1 0 = 0
- 16. Absorption Law Statement 1: A + AB = A Statement 2: A (A + B) = A Proof: Proof: A + AB = A.1 + AB → since A.1 = A A (A + B) = A.A + A.B =A(1+B) → since 1 + B = 1 = A+AB → since A . A = A = A.1 = A (1 + B) = A = A.1 = A
- 17. Redundancy laws Statement 3: Proof: A + ĀB = A + B A + ĀB = (A + Ā) (A + B) → since A+BC = (A+B)(A+C) using distributive law = 1 * (A + B) → since A + Ā = 1 =A + B Statement 4: Proof: A * (Ā+B) = AB A * (Ā + B) = A. Ā + AB = AB → since A Ā = 0
- 18. De Morgan’s Theorem Statement 1: “The negation of conjunction is the disjunction of the negations”. Or we can define that as “The compliment of the product of 2 variables is equal to the sum of the compliments of individual variables”. (A.B)’ = A’ + B’ Statement 2: “The negation of disjunction is the conjunction of the negations”. Or we can define that as “The compliment of the sum of two variables is equal to the product of the compliment of each variable”. (A + B)’ = A’.B’
- 19. De Morgan’s Theorem Verifying DeMorgan’s First Theorem using Truth Table Input Input Output Output Output Output Output B A A.B (A.B)’ A' B' A' + B' 0 0 0 1 1 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 0 0
- 20. De Morgan’s Theorem DeMorgan’s First Law Implementation using Logic Gates
- 21. De Morgan’s Theorem Verifying DeMorgan’s Second Theorem using Truth Table Input Input Output Output Output Output Output B A A+B A+B A B A . B 0 0 0 1 1 1 1 0 1 1 0 0 1 0 1 0 1 0 1 0 0 1 1 1 0 0 0 0
- 22. De Morgan’s Theorem DeMorgan’s Second Law Implementation using Logic Gates
- 23. DeMorgan’s Equivalent Gates Standard Logic Gate DeMorgan’s Equivalent Gate
- 24. DeMorgan’s Equivalent Gates Standard Logic Gate DeMorgan’s Equivalent Gate