Switching theory and logic design.
- 1. Switching Theory and Logic Design Boolean Algebra
- 2. Boolean Algebra A set B of elements (a,b,c,….) with an equivalence relation( =), two binary operations( + and .) and a unary operation( complementation – denoted by ‘ ) is a boolean algebra if and only if the following postulates are satisfied. Associativity Commutativity Distributivity Identity Complement
- 3. P.1 ) Associativity :- The + and . Operators are associative (a+b)+c = a+(b+c) (ab)c = a(bc) = abc P.2) Commutativity :- The + and . Operators are commutative a + b = b+ a ab = ba
- 4. P.3) Distributivity :- a + bc = (a+b)(a+c) a( b+c) = ab+ ac P.4) Identity Elements There exists an identity element (denoted by 0) for the + operation and another (denoted by 1) for the . Operation within B such that a+ 0 = a a 1=a
- 5. P.5) Complement Each member of B has a complement within B such that if a’ is the complement of a , then a + a’ = 1 aa‘ = 0
- 6. Fundamental Theorms on Boolean Algebra 1) Closure of Identity Elements For all a€B, a+1 = 1 and a0=0 2) Equality Theorem For all a,b,c € B , if a+b = a+c and ab= ac, then b = c
- 7. Complementary Theorem For all a,b €B, if a+b =1 and ab=0, then a = b’ and b=a’
- 8. DeMorgan’s Theorem (a+b)’ = a’b’ and (ab)’ = a’ + b’
- 9. Show That (A+B)(A+B’) = A
- 10. S T ABC + AB’C+ ABC’=A(B+C)
- 11. S T(A+B)(A+B’)(A’+C) =AC
- 12. S T AB+ A(B+C) + B( B+C) = B+ AC
- 13. S T (X + Y+XY)(X+Z) = (X+Y)(X+Z) ST (A+ B’+A’B)C’ = C’
- 14. Boolean Constants, Variables & Functions The Boolean algebra contains two constants 0 and 1 Any characters in lowercase or uppercase are used as Boolean Variables
- 15. Boolean Functions Any Boolean constants, variables, itself or combined with any of three operations(logical sum, logical product & complement) will give a Boolean expression. Any boolean expression can be written as a Boolean Function F = A’B + (AB)’ The above mentioned Boolean algebra is also called Switching algebra since the two constant values 0 and 1 represent off & on respectively in a circuit
- 16. Electronic Gates An electronic circuit that has one or more inputs and one output, and in which the electrical condition of the output at any time is dependent on those of the inputs at that time is called an electronic gate.
- 18. Compute the Boolean Function represented by the contact network
- 19. Calculate the switching functions realized by the contact network shown below
- 20. Boolean Functions & Logical Operations A Boolean variable in the true form or in the complemented form is called a literal. Eg: a,a’,b,b’ The Boolean product of two or more literals is called a product term Eg: ab’c , a’bc’ The Boolean sum of two or more literals is called a sum term Eg: (a+b’+c’) , (a’+b+c’)
- 21. Disjunctive Normal Form When a Boolean function appears as a sum of several product terms, it is said to be expressed as a sum of product(SOP) form. The SOP form is also called the Disjunctive normal form(DNF). Eg: f(abcd)=a’b+bc’+abcd
- 22. Conjunctive Normal Form When a Boolean function appears as a product of several sum terms, it is said to be expressed as a product of sums(POS). The POS form is also called the conjunctive normal form(CNF). F(abcd) = (a+b)(b’+c’+d)
- 23. Express the following function in DNF F = (a+b)(b+c+d) Express the following Boolean function in CNF F= ab+c’d F= abc+b’d
- 24. Canonical Forms When each of the terms of a Boolean function expressed either in SOP or POS form which has all the variables in it, it is said to be in canonical form. The canonical SOP form is called the disjunctive canonical form(DCF) and the canonical POS form is called the conjunctive canonical form(CCF)
- 25. Express the following function in a canonical form F = ab’+bc’+ac F= a’+bc Express the following function in a canonical form F = (a+b)(b+c’) F(a,b,c)= a+b
- 26. Fundamental Products and Sums A product term of n variables is called a minterm of n variables. Eg: ab’c’ , ab’c’ The no.of all possible minterms of n variables is 2n. A minterm is also called a fundamental product. A sum term of n variables is called a maxterm of n variables Eg: (a+b+c+d), (a+b’+c+d’) The no.of all possible maxterms of n variables is 2n. A maxterm is also called a fundamental sum.
- 29. Disjunctive canonical normal form(DCNF) of a Boolean function is a sum of minterms and the conjunctive canonical normal form(CCNF) is a product of maxterms. F=a’b’c’ + a’b’c + abc = m0+m1+m7 = ∑(0,1,7) F = (a+b+c)(a’+b’+c)(a’+b’+c’) = M0M6M7 = ∏(0,6,7) ∑ Indicates that the terms are minterms and the function is a summation ∏ Indicates that the terms are maxterms and the function is a product
- 30. Properties The Sum of all 2n minterms of n variables is 1. The product of any two n-variable minterms, which are different is 0. mimj=0 when i!=j Complement of mi is Mi mi’=Mi & Mi’=mi
- 31. Convert the given DCNF to CCNF F=a’bc + ab’c’ + abc Express the following function in both canonical forms F= a’+bc F(abcd) = (a+b)(b’+c’+d)
- 32. Theorem (Conversion b/w minterms & maxterms) A boolean function expressed as a sum of minterms or as a product of maxterms can be converted into the other forms as given by ∑mi = ∏Mj & ∏Mi=∑mj Where the subsets i and j are two partitions of the entire set of 2n subscripts of either m’s or M’s
- 33. Express the function f = a’c’d’ + a’bc’+ a’bcd’ in both types of canonical form Express the function f(a,b,c) = (a’+b+c)(a’+b’)(a+b+c) in both types of canonical form
- 34. Express the following function in CCF F(a,b,c) = ab’ +c’
- 35. Self Dual Function If a Boolean function and its dual are the same, then the function is called a self-dual function Check whether the given function is self dual or not f= ab+ac+bc Check whether f = b’(a’c’+ac) + b(a’c +ac’) is self dual or not
- 36. Logical Operations Like AND,OR etc. , there are three more significant boolean operators. They are NAND, NOR and EXCLUSIVE-OR
- 37. Properties of NAND operation SHOW That A↑B= B↑A Check whether NAND operation is associative or not
- 38. Properties of XOR operation XOR is commutative XOR is also associative
- 39. Karnaugh Maps(Veitch Diagram) The number of gates required and the number of input terminals for the gates for the implementation of a Boolean function get reduced considerably if the Boolean function can be simplified. Karnaugh Map method gives a systematic approach for simplifying a Boolean expression.
- 40. K-map contains boxes called cells. Each of the cell represents one of the 2n possible products(of the variables) that can be formed from n variables. Thus a 3-variable map contains 8 cells, 2 variable-4 cells, 4 variable-16 cells. Each nearby cells represents only one bit change. Numbering of each cells will be done based on the gray code sequence.
- 44. Plotting SOP form in K-map If a Boolean function is given as a sum of its minterms(ie in its DCF form), then the function is plotted on the map by writing 1’s on those cells which represent the minterms that are present in the function.
- 45. Plot the Boolean function Y = ABC’ + ABC + A’B’C on the K-map Plot the boolean function Y = A’BC’D’ + AB’C Plot the boolean function Y = AB + BC Plot f(abc) = ∑(0,2,6,7) in the K-map Plot f(abcd) = ∑(1,3,4,7-10,12,15)
- 46. Plotting POS form in K-map If a Boolean function is given as a product of maxterms (ie in CCF), then the function is plotted on the map by writing 0’s on those cells which represent the maxterms that are present in the function.
- 49. Represent f = ∏(0,3,6,7) Plot function f(a,b,c)= a +b in K-map Plot f= c’d’ + a’bd Plot f = b’c + d + abc’
- 50. K-map minimization Two cells are said to be adjacent if there is only one bit change among the cells. Group those cells and eliminate variables which is in true form and complementary form.
- 61. Minimum Sum Of Products Minimize the expression Y = A’B’C’+A’BC+AB’C’+ABC Y= AB’C + A’B’C + A’BC + AB’C’ + A’B’C’
- 62. Minimize the expression Y= A’BC’D’ + A’BC’D + ABC’D’ + ABC’D + AB’C’D + A’B’CD’
- 63. Minimum Product of Sums Minimize the expression Y = (A+B+C’) (A+B’+C’) (A’+B’+C’)(A’+B+C) (A+B+C) Write each Maxterms. Plot 0 for the Maxterm cells
- 64. Minimize the expression F= ∏(0,2,3,8,9,12,13,15)
- 65. 5-variable K-map Simplify F = ∑(0,2,4,6,9,11,13,17,21,25,27,29,31)
- 66. Don’t care terms The combinations for which the values of the expression are not specified are called don’t care combinations. The output is a don’t care term for these invalid combinations
- 67. Minimize the boolean expression of the below circuit logic A B C Y 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1
- 68. Prime Cubes Each cells in the k-map corresponds to minterms is called a cube. A cube of a given function that can not grow larger by expanding into other cubes of the function is called a prime cube(PC) or a prime implicant (PI) of the function.
- 69. Find the prime implicants/ prime cubes of the function f = ∑(2,5,6)
- 70. Essential Prime Cubes If among the minterms subsuming a prime cube, there is at least one that is covered by this and only this prime cube, then the prime cube is called an essential prime cube (EPC) or an essential prime implicant(EPI)
- 71. Redundant Prime Cubes If each of the minterms subsuming a prime cube is covered by other essential prime cubes, then the prime cube is a redundant prime cube(RPC) or a redundant prime implicant Find the EPC and RPC of the following boolean function ∑(1,3,6,7)
- 72. Quine-Mc Cluskey Minimization wrt to minimization, K-maps Only good for: Small functions (<6variables) Single output function at a time Not Implementable on Computer Subjective Interpretation, Different Coverings Q-M (Tabular Minimization) Solves These Problems
- 73. Basic Definitions Any single or group of ones that can be combined is called an implicant. A prime implicant of a function F is a product term implicant which is no longer an implicant if any literal is deleted from it. Essential prime implicants are the implicant which will definitely in the final expression.
- 74. Procedure F (w,x,y,z)= (0,1,2,3,5,7,8,10,14,15) Write the binary equivalent of all the minterms. Group all the minterms by the no of 1s they contain.
- 75. Step 1 0 0000 1 0001 2 0010 8 1000 3 0011 5 0101 10 1010 7 0111 14 1110 15 1111
- 76. Combine the members of new groups to create more new groups. Combined group must differ by one bit and must have – in the same position. Combine as much as possible. Select prime implicants to cover all the ones.
- 77. Finding Prime Implicants 0 0000 1 0001 2 0010 8 1000 3 0011 5 0101 10 1010 7 0111 14 1110 15 1111 Step 1 Step 2 (0,1) 000- (0,2) 00-0 (0,8) -000 (1,3) 00-1 (1,5) 0-01 (2,3) 001- (2,10) -010 (8,10) 10-0 (3,7) 0-11 (5,7) 01-1 (10,14) 1-10 (7,15) -111 (14,15) 111- Step 3 (0,1,2,3) 00-- (0,2,1,3) 00-- (0,2,8,10) -0-0 (0,8,2,10) -0-0 (1,3,5,7) 0--1 (1,5,3,7) 0--1 6 Prime Implicants 1-10 -111 111- 00-- -0-0 0--1 (0,1) 000- (0,8) -000 (2,3) 001-
- 78. Finding Prime Implicants 0 0000 1 0001 2 0010 8 1000 3 0011 5 0101 10 1010 7 0111 14 1110 15 1111 Step 1 Step 2 (0,1) 000- (0,2) 00-0 (0,8) -000 (1,3) 00-1 (1,5) 0-01 (2,3) 001- (2,10) -010 (8,10) 10-0 (3,7) 0-11 (5,7) 01-1 (10,14) 1-10 (7,15) -111 (14,15) 111- Step 3 (0,1,2,3) 00-- (0,2,1,3) 00-- (0,2,8,10) -0-0 (0,8,2,10) -0-0 (1,3,5,7) 0--1 (1,5,3,7) 0--1 6 Prime Implicants 1-10 -111 111- 00-- -0-0 0--1 (0,1) 000- (0,8) -000 (2,3) 001-
- 79. Find Essential Prime Implicants – Prime Cube table Prime Implicant Covered minterms 1-10 -111 111- 00-- -0-0 0--1 Minterms 0 1 2 3 5 7 8 10 14 15 10,14 7,15 14,15 0,1,2,3 0,2,8,10 1,3,5,7 X X X X X X X X X X X X X X X X X X Essential Prime implicant Here essential prime cubes are x’z’ & w’z
- 80. Selective Prime Cube Table • In an SPC table, If a row I has crosses in all columns where another row J has crosses, and there is at least one more column with a cross under I but not under J, row I is said to dominate row J • A row in an SPC table that is dominated by another row of the same or smaller cost can be deleted from the SPC table
- 81. 14 15 A(10,14) * B(7,15) * C(14,15) * * • As per the theorem , Here C can be selected.