1. Name the person who developed Boolean algebra
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- Lorraine Johns
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1 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES Name the person who developed Boolean algebra Boolean algebra is a logical calculus of truth values, developed by George Boole in the late 83s. The laws of Boolean algebra can be defined axiomatically as certain equations called axioms together with their logical consequences called theorems 2. What is a logic gate? Name three logic gates Boolean logic is simply a way of comparing individual bits. It uses what are called operators to determine how the bits are compared. They simulate the gates like AND, OR and NOT 3. What is a truth table? The output will be off if any of the inputs are off. The OR operation says if any input is on, the output will be on. It's easy to see all of the combinations by using what are called truth tables, illustrated below. At the bottom of each table is shown the schematic symbol found in circuit diagrams. AND (all high = high, else low) Input Input Output 2 OR (any high = high, else low) Input Input Output 2
2 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES State the answer for the following a.a+=? b.a.=? Answer: A+ =A A.= 5. What do you understand by maxterm and minterm. A Boolean function is in canonical form when it is expressed as a sum of minterms or as a product of maxterms. A minterm is defined as a Boolean function that is equal to in only one row of its truth table and in all other rows. Where as a maxterm is a Boolean function that is equal to in only row of its truth table and in all other rows. 6. What are the types of Boolean expression? Boolean functions can be expressed in three different ways: canonical form, standard form, and nonstandard form. 7. Why is NAND gate called a universal gate? NAND is called a universal gate because we can represent the other basic gates using NAND gates. 8. What is k-map? A Karnaugh map provides a pictorial method of grouping together expressions with common factors and therefore eliminating unwanted variables. The Karnaugh map can also be described as a special arrangement of a truth table. 9. How many variable are reduced by a pair, quad and octet respectively? Two variables are reduced by a pair, four variables are reduced by a quad and eight variables by octet.. What do you understand by logical function? The operators used most often are AND and OR. The AND operation says if and only if all inputs are on, the output will be on. The output will be off if any of the inputs are off.. What are the basic postulates of Boolean algebra? 2
3 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES After values, the next ingredient of any algebraic system is its operations. Whereas elementary algebra is based on numeric operations such as multiplication xy, addition x + y, and negation x. Conjunction is the closest to these three to its numerical counterpart, in fact on and it ismultiplication. As a logical operation the conjunction of two propositions is true when both propositions are true, and otherwise is false. The first column of Figure below tabulates the values of x y for the four possible valuations for x and y; such a tabulation is traditionally called a truth table. Disjunction, in the second column of the figures, works almost like addition, with one exception: the disjunction of and is neither 2 nor but. Thus the disjunction of two propositions is false when both propositions are false, and otherwise is true. This is just the definition of conjunction with true and false interchanged everywhere; because of this we say that disjunction is the dual of conjunction. Logical negation however does not work like numerical negation at all. Instead it corresponds to incrementation: x = x+ mod 2. Yet it shares in common with numerical negation the property that applying it twice returns the original value: x = x, just as ( x) = x. An operation with this property is called an involution. The set {, } has two permutations, both involuntary, namely the identity, no movement, corresponding to numerical negation mod 2 (since + = mod 2), and SWAP, corresponding to logical negation. Using negation we can formalize the notion that conjunction is dual to disjunction via De Morgan's laws, (x y) = x y and (x y) = x y. These can also be construed as definitions of conjunction in terms of disjunction and vice versa: x y = ( x y) and x y = ( x y). 2. Explain about logical operations. The operators used most often are AND and OR. The AND operation says if and only if all inputs are on, the output will be on. The output will be off if any of the inputs are off. The OR operation says if any input is on, the output will be on. It's easy to see all of the combinations by using what are called truth tables, illustrated below. At the bottom of each table is shown the schematic symbol found in circuit diagrams. AND (all high = high, else low) Input Input 2 Output OR (any high = high, else low) Input Input Output 3
4 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES There are two operations that have the same logic as above, but with an inverted output. The NAND operation says if and only if all inputs are on, the output will be off. The output will be on if any of the inputs are off. The NOR operation says if any input is on, the output will be off. Notice the bubble on the output of the schematic symbol used to indicate an inversion. NAND (all high = low, else high) Input Input 2 Output NOR (any high = low, else high) Input Input 2 Output There is a variation on the OR logic called Exclusive OR or XOR. Exclusive OR says the output will be on if the inputs are different. Another one, the inverter or NOT operation, says that the output will be opposite in state to the input. It obviously has only one input and one output. Note that it will change an AND to a NAND, an OR to a NOR and an XOR to a NXOR if connected to their outputs. It simply changes s to s and s 4
5 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES to s. XOR (different = high, same = low) Input Input 2 Output NOT (inverter) Input = Output = Input = Output = 3. Explain Logic gates in detail The basic operations used to combine propositions are as follows: And: Let P and Q be two propositions, P and Q is true if and only if both P and Q are true. P and Q can also be represented as P * Q but still it is pronounced as "and", and it is also called the "conjunction" of P and Q. = True =False P Q P * Q Or: Let P and Q be two propositions, P or Q is true when even one of these two i.e. P or Q is true. P and Q can also be represented as P + Q but still it is pronounced as "or", and it is also called the "disjunction" of P or Q. P Q P + Q 5
6 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES = True = False Note: Let P be a proposition, the compound proposition not P, is true if P is false. Not P can also be represented as ~ P but still it is pronounced as "not P", and it is also called the "negation" of P P ~P = True 4. Explain in detail about basic theorem of Boolean algebra The following shows the basic Boolean laws. Note that every law has two expressions, (a) and (b). This is known as duality. These are obtained by changing every AND (.) to OR (+), every OR (+) to AND (.) and all 's to 's and vice-versa. It has become conventional to drop the. (AND symbol) i.e. A.B is written as AB. T : Commutative Law (a) A + B = B + A (b) A B = B A T2 : 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 = A T5 : (a) 6
7 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES (b) T6 : Redundance Law (a) A + A B = A (b) A (A + B) = A T7 : (a) + A = A (b) A = T8 : (a) + A = (b) A = A T9 : (a) (b) T : (a) (b) T : De Morgan's Theorem (a) (b) 5. Expand using algebraic and truth table (a) () Algebraically: 7
8 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES (2) Using the truth table: Using the laws given above, complicated expressions can be simplified. 6. How will you derive an Boolean expression. Boolean functions can be expressed in three different ways: canonical form, standard form, and nonstandard form. The canonical form has the advantage of being unique; although it will not use the minimal number of operators for a given Boolean function. In comparison with this canonical form, the standard form tends to require fewer operators, and the nonstandard form tends to use the smallest number of the three. Each of the two forms can be obtained from the canonical form through algebraic manipulation. In this section of the tutorial, you will be taught how to express Boolean functions in the canonical form. A Boolean function is in canonical form when it is expressed as a sum of minterms or as a product of maxterms. A minterm is defined as a Boolean function that is equal to in only one row of its truth table and in all other rows. Whereas a maxterm is a Boolean function that is equal to in only row of its truth table and in all other rows. Here are the following steps to convert an algebraic expression into a sum-of-minterms, using a truth table. ) Derive a truth table from the algebraic expression. 2) Read the minterms from the truth table. A minterm is identified by a row where F =. 3) Collect all the minterms and combine them into a sum-of-minterms. 7. State an example for solving using minterm and maxterm. Sum-of-minterms Converting the algebraic expression, F = x +yz, into a sum-of-minterms. ) Derive a truth table from the algebraic expression, F = x + zy. 8
9 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES X Y Z F (m) (m) (m2) (m3) (m4) (m5) (m6) 2) Read the minterms from the truth table. The minterms are: m3, m4, m5, and m6 3) Collect all the minterms and combine them into a sum-of-minterm. The sum-of-minterm is expressed as follow: F = m3 + m4 + m5 + m6 OR F = ~xyz + x~y~z + x~yz + xyz The minterm ~xyz is derived by examining row m3 in the truth table. In m3 we can see that x =, y =, and z =. When one of the variable (ie. x,y,or z) are equal to we represent it by ~x and when it is equal to we represent it as x. Thus, we get the following minterms ~xyz, x~y~z, and xyz. To convert an algebraic expression into a product-of-maxterms, using a truth table, is similar to converting to a sum-of-minterms. Here are the following steps: ) Derive a truth table from the algebraic expression. 2) Read the maxterms from the truth table. A maxterm is identified by a row where F =. 3) Collect all the maxterms and combine them to get the product-of-maxterms. Example: product-of-maxterms Converting the algebraic expression, F = ~x~y + xz, into a product-of-maxterms. ) Derive a truth table from the algebraic expression, F = ~x~y + xz. X Y Z F (m) (m) (m2) (m3) (m4) (m5) (m6) (m7) 2) Read the maxterms from the truth table. The maxterms are: m2, m3, m4, and m6. 3) Collect all the maxterms and combine them to get the product-of-maxterms. The product-ofmaxterms is expressed as follow: F = m2m3m4m6 9
10 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES OR F = (~x + y + ~z) (~x + y + z) (x + ~y + ~z) (x + y + ~z) 8. What are the various types of simplification of Boolean algebra. Karnaugh Maps K-Maps provides a simple and straight-forward method of minimizing Boolean expressions. With the Karnaugh map Boolean expressions having up to four and even six variables can be simplified. Tabular Method The tabular method is a tedious and lengthy process for human designers but quite suitable for machine computation, because it searches exhaustively for prime implicants and eventually finds all possible covers. Algebraic Method The algebraic method can transform one Boolean expression into an equivalent expression by applying Boolean Theorems. This is important if you want to convert a given expression to a standardized form or if you want to minimize the number of literals or terms in an expression. 9. How do you simplify an expression using k-map The Karnaugh map uses the following rules for the simplification of expressions by grouping together adjacent cells containing ones The following rules are for evaluating the minterm or SOP (sum of product) and the same rules are held for maxterm or POS (product of sum) but the only difference is in minterm we consider (mark) for the cells containing one but for maxterm we concern (mark) for the cells containing zero. Groups may not include any cell containing a zero (for minterm). Groups may be horizontal or vertical, but not diagonal.
11 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES Groups must contain, 2, 4, 8, or in general 2n cells. That is if n =, a group will contain two 's since 2 = 2. If n = 2, a group will contain four 's, 2 2 = 4. Each group should be as large as possible. Each cell containing a one must be in at least one group.
12 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES Groups may overlap. Groups may wrap around the table. The leftmost cell in a row may be grouped with the rightmost cell and the top cell in a column may be grouped with the bottom cell. There should be as few groups as possible, as long as this does not contradict any of the previous rules. 2
13 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES What are the rules to be considered for k-map simplification?. No zeros allowed. 2. No diagonals. 3. Only power of 2 numbers of cells in each group. 4. Groups should be as large as possible. 5. Every one must be in at least one group. 6. Overlapping allowed. 7. Wrap around allowed. 8. Fewest numbers of groups possible. 2. Consider the following map. The function plotted is: Z = f (A,B) = A + AB ) Note that values of the input variables form the rows and columns. That is the logic values of the variables A and B (with one denoting true form and zero denoting false form) form the head of the rows and columns respectively. 2) Have in mind that the above map is a one-dimensional type, which can be used to simplify an expression in two variables. There is a two-dimensional map that can be used for up to four 3
14 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES variables, and a three-dimensional map for up to six variables. The result: Variable B becomes redundant due to ~A + A = or ~A*A = 22. Explain in detail the NAND, NOR and XOR gate Now that we have learnt the basic operations of Boolean Algebra, we can proceed into more advanced Boolean Operations. In this lesson, we will introduce you to more complex, advanced Boolean operations. NAND The NAND operation, in other words NOT-AND, is equivalent to an AND operation followed by a NOT operation. The output of the NAND operation is false if and only if both of its inputs are true. We can also think of this operation as opposite of AND operation, and hence can be pronounced as "not and" the truth table for NAND is given below: P Q P NAND Q = True = False NOR The NOR operation, in other words NOT-OR, is equivalent to an OR operation followed by a NOT operation. The output of the NOR operation is false if any of its inputs are true. We can also think of this operation as opposite of OR operation, and hence can be pronounced as "not or" the truth table for NOR is given below: P Q P NOR Q = True = False XOR Let P and Q are two propositions, P XOR Q is true when only one of these two i.e. either P or Q is true, but not both. It is represented as P XOR Q and it is pronounced as "X-OR" or "exclusive OR". The truth table for XOR is given below: P Q P XOR Q 4
15 MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES = True = False 5
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