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### Computer science study material

1. 1. Page 1 of 10 BOOLEAN ALGEBRA1 P0INTS TO REMEMBER1 .BOOLEAN ALGEBRA: Is a two state algebra or algebra of logics.It is also calledSwitching Algebra. It is based on Binary number system and uses the numeric constants0 and 1.2. BINARY DECISION: The decision which results into either ‘ TRUE ‘ or ‘FALSE’where TRUE stands for 1 and FALSE stands for 0.3. LOGICAL VARIABLES(Boolean Variable or Binary valued Variable):Thevariables which can stores values either TRUE or FALSE OR ‘0’ or ‘1’.4 BINARY OPERATIONS :Is an operation in which for a set of variables result is thevalues i.e 0 or 1.5 BOOLEAN OPERATORS:: In Boolean Operation ,operators used are of three types: a) NOT- It is a Unary Operator.i.e it operateson single variable and operation performed by it is known as Complementation or Negation .Its symbol is “¯”or “ ’ ”.e.g. A’ or Ā b)AND – It is a Binary Operator . It operates on two variables and operation . performed by it is known as Logical Multiplication . Its Symbol is “ ” Or “ ˆ ”. . e.g. A B or Aˆ B. c) OR- It is a Binary Operator . It operates on two variables and operation performed by it is known as Logical Addition . Its symbol is “ +” or “ˇ”. e.g. . A + B or A ˇ B. 6. TRUTH TABLE – A truth table is a table that describes the behaviour of a logic gate.It shows all input and output possibilities for logical variables or statements.The input patterns are written in Binary Progression. 7.TAUTOLOGY- If the result of a logical statement or exprssion is always true or ‘1’, it is known as Tautology. 8. FALLACY-- If the result of a logical statement or exprssion is always False or ‘0’, it is known as Fallacy. 9 .LOGIC GATES-A logic gate performs a logical operation on one or more logic inputs and produces a single logic output. Boolean algebraType Distinctive shape Truth table between A & B INPUT OUTPUT A B A AND BAND A ..B 0 0 0 0 1 0 Prepared By Sumit Kumar Gupta, PGT Computer Science
2. 2. Page 2 of 10 1 0 0 1 1 1 INPUT OUTPUT A B A OR B 0 0 0OR A+B 0 1 1 1 0 1 1 1 1 INPUT OUTPUT Ā A NOT ANOT 0 1 1 0 INPUT OUTPUT A NAND A B BNAND 0 0 1 0 1 1 1 0 1 1 1 0 Prepared By Sumit Kumar Gupta, PGT Computer Science
3. 3. Page 3 of 10 INPUT OUTPUT A B A NOR B 0 0 1NOR 0 1 0 1 0 0 1 1 0 INPUT OUTPUT A B A XOR B 0 0 0XOR 0 1 1 1 0 1 1 1 0 INPUT OUTPUT A XNOR A B B 0 0 1XNOR 0 1 0 1 0 0 1 1 1BOOLEAN POSTULATES  P1: X = 0 or X = 1  P2: 0 . 0 = 0  P3: 1 + 1 = 1  P4: 0 + 0 = 0  P5: 1 . 1 = 1  P6: 1 . 0 = 0 . 1 = 0  P7: 1 + 0 = 0 + 1 = 1 Prepared By Sumit Kumar Gupta, PGT Computer Science
4. 4. Page 4 of 10LAWS OF BOOLEAN ALGEBRAT1 : Commutative Law (a) A + B = B + A (b) A B = B AT2 : Associate Law (a) (A + B) + C = A + (B + C) (b) (A B) C = A (B C)T3 : Distributive Law (a) A (B + C) = A B + A C (b) A + (B C) = (A + B) (A + C)T4 : Identity Law (a) A + A = A (b) A A = AT5 : (a) (b)T6 : Redundance Law (a) A + A B = A (b) A (A + B) = AT7 : (a) 0 + A = A (b) 0 A = 0T8 : (a) 1 + A = 1 (b) 1 A = AT9 : (a) (b)T10 : (a) (b)T11 : De Morgans Theorem (a) (b) ExamplesProve T10 : (a) Prepared By Sumit Kumar Gupta, PGT Computer Science
5. 5. Page 5 of 10S(1) Algebraically:(2) Using the truth table:Using the laws given above, complicated expressions can be simplified. Problems(a) Prove T10(b).(b) Copy or print out the truth table below and use it to prove T11: (a) and (b). Prepared By Sumit Kumar Gupta, PGT Computer Science
6. 6. Page 6 of 10(b) DEMORGAN THEOREM(a) Proof of (A + B) = A . B,These can be proved by the use of truth tables 1(a) & 1(b) table 1(a)A B A+B (A+B)0 0 0 10 1 1 01 0 1 01 1 1 0 table 1(b)A B A B A.B0 0 1 1 10 1 1 0 01 0 0 1 01 1 0 0 0The two truth tables are identical, and so the two expressions are identical.(b)Proof of (A.B) = A + B(A.B) = A + B, These can be proved by the use of truth tables 2(a) & 2(b) table 2(a)A B A.B (A.B)0 0 0 1 Prepared By Sumit Kumar Gupta, PGT Computer Science
7. 7. Page 7 of 100 1 0 11 0 0 11 1 1 0 table2(b)A B A B A+B0 0 1 1 10 1 1 0 11 0 0 1 11 1 0 0 0Canonical form: standard form for a Boolean expressionprovides a unique algebraic signatureMinterms and MaxtermsAny boolean expression may be expressed in terms of either minterms ormaxterms.A literal is a single variable within a term which may or may not becomplemented. For an expression with N variables, minterms and maxterms aredefined as follows :A minterm is the product of N distinct literals where each literal occurs exactlyonceA maxterm is the sum of N distinct literals where each literal occurs exactly onceFor a two-variable expression, the minterms and maxterms are as followsX Y Minterm Maxterm0 0 X.Y X+Y0 1 X.Y X+Y1 0 X.Y X+Y1 1 X.Y X+YFor a three-variable expression, the minterms and maxterms are as follows Prepared By Sumit Kumar Gupta, PGT Computer Science
8. 8. Page 8 of 10X Y Z Minterm Maxterm0 0 0 X.Y.Z X+Y+Z0 0 1 X.Y.Z X+Y+Z0 1 0 X.Y.Z X+Y+Z0 1 1 X.Y.Z X+Y+Z1 0 0 X.Y.Z X+Y+Z1 0 1 X.Y.Z X+Y+Z1 1 0 X.Y.Z X+Y+Z1 1 1 X.Y.Z X+Y+ZSum Of Products (SOP)The Sum of Products form represents an expression as a sum ofminterms.To derive the Sum of Products form from a truth table, OR together allof the minterms which give a value of 1.Example – SOPConsider the truth tableX Y F Minterm0 0 0 X.Y0 1 0 XY1 0 1 X.Y1 1 1 X.YHere SOP is f(X.Y) = X.Y + X.YProduct Of Sum (POS)The Product of Sums form represents an expression as a product of maxterms.To derive the Product of Sums form from a truth table, AND together all of themaxterms which give a value of 0.Example – POSConsider the truth table from the previous example.X Y F Maxterm0 0 1 X+Y0 1 0 X+Y Prepared By Sumit Kumar Gupta, PGT Computer Science
9. 9. Page 9 of 101 0 1 X+Y1 1 1 X+YHere POS is F(X,Y) = (X+Y)Minimisation of Boolean FunctionsIn mathematics expressions are simplified to understand and easier to writedown, they are also less prone to error.Minimisation can be achieved by a following methods:1)Algebraic Manipulation of Boolean Expressions.2)Karnaugh MapsAlgebraic Manipulation of Boolean ExpressionsThis is an approach where you can transform one boolean expression into anequivalent expression by applying Boolean TheoremsKarnaugh Maps  K-Maps are a convenient way to simplify Boolean Expressions.  They can be used for up to 4 or 5 variables.  They are a visual representation of a truth table.  Expression are most commonly expressed in sum of products form. . Prepared By Sumit Kumar Gupta, PGT Computer Science
10. 10. Page 10 of 10Prepared By Sumit Kumar Gupta, PGT Computer Science