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- 2. PREFACE • Here, Iam presenting a project on the topic “Boolean algebra” In this project I have tried to give all the important things about the project. • I am thankful to my computer teacher who gave me moral support and guided me to complete this project on time. He also guided me in research work for the project. While doing this project I came across many new thing which improved my Skill. • I have given all the information in this project by consulting subject specialist, teachers, books and websites
- 3. CONTENT • Introduction • Propositionallogic • Terms related to connectives • Type of Connective • Converse,Inverse,Contrapositive • Logical operator • Boolean law • Principal of duality • Karnaugh map • Two variables k map • Three variables k map • Four variables k map • Conclusion • Bibliography
- 4. INTRODUCTION • Logic isa collection of rules for reasoning. Logical reasoning has been observed over the years and the emerging patterns have been extracted and streamlined. In fact, the foundation of logical evolution was laid by a British Mathematician George Boole in the middle of 19th century. • In logic, we discuss about the 'true' or 'false' value of the statements and learn how to determine it with the help of other statements. To promote clarity of thought and to eliminate errors, symbols are used rather than specific statements.
- 5. PROPOSITIONAL LOGIC • Propositionallogic, also known as sentential logic and statement logic, is the branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences, as well as the logical relationships and properties that are derived .
- 6. TERMS RELATED TOCONNECTIVES • Common connectives include negation, disjunction, conjunction, implication, and equivalence. In standard systems of classical logic, these connectives are interpreted as truth functions, though they receive a variety of alternative interpretations in non classical logics.
- 7. TYPES OF CONNECTIVES •Disjunction ('v' or '+'): Disjunction is the term used to combine two or more propositional statements with 'or' connective. It is represented with the symbol 'v' or '+'. It results in 'False', if both the operands are False otherwise, "True'.
- 8. • Conjunction (‘^’or ‘ , ’ ) : Conjunction is the term used to join two or more propositional statements with the help of 'and' connective. It is represented by symbol '^' or '.'(dot). The conjunction of two or more propositions results in 'True', if all the propositions are 'True' otherwise, 'False'.
- 9. • Negation (~or '): Negation of a propositional statement is formed by using symbol 'not' with the verb of the sentence. It is represented as ( ~a ) or (a'). • Double negation (~~ or "): Double negation means complement of complement or double complement which results in the same truth value. This means that: ~(~a) = a or (a')' = a .
- 10. • Implication/conditional connective(): when two propositional statements are joined together such that the second statement is a logical consequent of the first, the relation is said to be an implication. If A and B are two propositions, then the implication of A on B can be denoted by the symbol . here, A B refers to as 'A implies B' and has its literal meaning as 'if A is true then B is true'. It can also be represented as (a' + b).
- 11. • Equivalence/Bi-conditional (<->):It is a connective that joins two logical statements in such a way that the result is true, if both of them have same truth values. It is called an ‘ if and only if ’ connective and is represented by the symbol <->. If A and B are two propositions then A <-> B refers to as 'A bi- conditional B'.
- 12. CONVERSE • Converse :The Converse Of Conditional Proposition Is Determine By Interchanging The Antecedent And Consequent Of Given Conditional Proposition Is Known As Converse. • ‘If A Then B’ Will Be Written As ‘If B Then A’.
- 13. INVERSE • Inverse :The Inverse Of Conditional Propositional Another Conditional Having Negated Antecedent Consequent Of Given Conditional Proposition Is Known As Inverse. ‘If A then B’ will be written as ‘if ~A then ~B’.
- 14. CONTRAPOSITIVE • Contrapositive :The Contrapositive Of A Conditional Is Formed By Creating Another Conditional That Takes It Antecedent And Negated Consequent Of Earlier Conditional . ‘If A then B’ will be written as ‘if ~B then ~A’.
- 15. TAUTOLOGY • Tautology :the preposition having 1’s in there truth table column . a 1 av1 0 1 1 1 1 1
- 16. CONTRADICTION • CONTRADICTION :THE PREPOSITION HAVING NOTHING BUT 0’s IN THERE TRUTH TABLE COLUMN ARE CALLED CONTRADICTION. a 0 av0 0 0 0 1 0 0
- 17. CONTINGENCY • Contingency :The Preposition That Having Some Combination Of 1’s And 0’s In Truth Table Column Are Called Contingency . a b avb a^(avb) 0 0 0 0 0 1 1 0 1 0 1 1 1 1 1 1
- 18. LOGICAL OPERATORS • LOGICALNOT : This operator provides the complement of the given binary valued quantity i.e, it converts 0’s into 1’s and 1’s into 0’s. It is represented as (‘) or (-)
- 19. • Logical or: this operator results in logical addition of two or more binary valued quantities. It is demoted by a plus (+) sign.
- 20. • Logical and: The AND operator results in logical product of two or more binary valued quantities. It is denoted by a dot (.)sign.
- 21. Compound Boolean expression • logicalnor : Logical NOR expression is formed by using a combination of OR and NOT operators.
- 22. • Logical NAND: Logical NAND expression is formed by using a combination of AND and NOT operators.
- 23. • Logical XOR: This operation in the short form of Exclusive-OR or Ex-OR. The logical XOR expression is formed by using a combination all the three fundamental operators AND ,OR and NOT.
- 24. • Logical XNOR: XNOR operation is opposite to XOR operation. It is a combination of XOR and NOT operators.
- 25. BOOLEAN POSTULATES AND THEOREM Apostulate is a statement that is assumed to be True for the basis of further reasoning or discussion. Postulates are also called ‘axioms. They do not need to be proved but they can be used to prove the theorems.
- 26. Properties of 0and 1: The value of any Boolean variable may either be 0 or 1. Hence, 1 added to the variable results in 1 and 0 multiplied to a variable results in 0. A+1=1 A•0=0
- 27. • Identity Law:This law states that 0 added to a variable and 1 multiplied to a variable results in the same value. • A+0=A • A•1=A
- 28. Complementary law: Thislaw states that a variable added to its complement results in 1 and the variable multiplied with its complement results in 0. • A+ A’=1 • A•A’=0
- 29. • Idempotent law:This law states that when a variable is either added or multiplied to the same variable, it will result in the same variable. • A+A=A • A•A=A
- 30. • Involution law:This law states that the double complement of any variable results in the same variable. • (A’)’=A
- 31. Commutative law: Thislaw states that a finite sum or product among variables, remains unchanged, by reordering the variables. • A+B=B+A • A.B=B.A
- 32. • Distributive law: This law states that • First law : A(B+C)=A•B+A•C • Second law : A+BC=(A+B)•(A+C)
- 33. Associative law :This law allows the removal of brackets from an expression and the regrouping of the variables, such that the output/result remains unchanged. • First law: A+ (B+C) = (A + B) + C • • Second law: A• (B•C) = (A•B)•C
- 34. Absorption law: Absorptionlaw states that: First law: A + A•B =A • Second law: Α•(A + B) = A
- 35. De Morgan’s Theorem •Augustus De Morgan, a great mathematician, developed two theorems which have brought immense utilities in Boolean algebra. • First theorem : (A+B)’=A’•B’ • Second theorem : (A•B)’=A’+B’
- 36. PRINCIPLE OF DUALITY •A Boolean expression can be converted into another form by replacing each plus (+) sign with a dot (.) and each dot (.) sign with a plus (+), each 1 with 0 and each 0 with 1. The variables must remain the same during the process. The expression so obtained, is known as the dual of the given Boolean expression and the process of conversion is termed as the Principle of Duality.
- 37. KARNAUGH MAP • Youhave come across simplification of boolean expression by using postulates or laws. This is possible only when the expression is in short form and can be simplified by using limited steps. Sometimes, it becomes very difficult to simplify an expression when its derivation is too long. • Maurice Karnaugh, a great mathematician, developed a way to simplify an expression by using a tabular or matrix representation, that enables you to obtain an expression in its most simplified form. This tabular representation is termed as K-map or Karnaugh map. • The advantage of using K-map is that the expression can be reduced to its most simplified form without any algebraic derivation. However, K-map has a limitation that it will reduce an expression only when it is in canonical SOP or POS form.
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- 41. Conclusion Boolean algebra providesa framework for working with binary variables and logical operations, essential in computer science and digital circuit design. It involves operations such as AND, OR, and NOT, governed by laws like commutative, associative, and distributive. Key concepts include simplification of expressions using laws and theorems like De Morgan’s laws. Boolean algebra’s principles enable the design of efficient digital systems and logic circuits, facilitating error correction and optimization. Overall, it serves as a foundational tool for analysing and designing complex logical structures in technology and computing.
- 42. Bibliography • From :-ISC CLASS XII UNDERSTANDING ISC COMPUTER SCIENCE