Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Company LOGO DKT 122/3 DIGITAL SYSTEM 1 WEEK #5 BOOLEAN ALGEBRA (PART 1) Boolean Algebra Contents Boolean Operations & Expression Laws & Rules of Boolean algebra DeMorgan’s Theorems Boolean analysis of logic circuits Simplification using Boolean Algebra Standard forms of Boolean Expressions Boolean Expressions & truth tables Boolean Algebra (Cont.) The Karnaugh Map Karnaugh Map SOP minimization Karnaugh Map POS minimization Programmable Logic Boolean Operations & Expression Expression Variable a symbol used to represent logical quantities (1 or 0) E.g : A, B,..used as variable Complement inverse of variable and is indicated by bar over variable E.g : Ā Boolean Operations & Expression Operation Boolean Addition – equivalent to the OR operation A B X X=A+B Boolean Multiplication – equivalent to the AND operation A X = A∙B B X Laws of Boolean Algebra Commutative Addition & multiplication Associative Addition & multiplication Distributive Same as ordinary algebra Commutative Law Addition Commutative law of addition: A+B = B+A the order of OR-ing does not matter. Commutative Law Multiplication Commutative law of Multiplication AB = BA the order of ANDing does not matter. Associative Law Addition Associative law of addition A + (B + C) = (A + B) + C The grouping of ORed variables does not matter Associative Law Multiplication Associative law of multiplication A(BC) = (AB)C The grouping of ANDed variables does not matter Distributive Law A(B + C) = AB + AC Question: (A+B)(C+D) ? Rules of Boolean Algebra Rule 1: A + 0 = A In math if you add 0 you have changed nothing. In Boolean Algebra ORing with 0 changes nothing. Rules of Boolean Algebra Rule 2: A + 1 = 1 ORing with 1 must give a 1 since if any input is 1 an OR gate will give a 1. Rules of Boolean Algebra Rule 3: A . 0 = 0 In math if 0 is multiplied with anything you get 0. If you AND anything with 0 you get 0. Rules of Boolean Algebra Rule 4: A . 1 = A ANDing anything with 1 will yield the anything. Rules of Boolean Algebra Rule 5: A + A = A ORing with itself will give the same result Rules of Boolean Algebra Rule 6: A + A = 1 Either A or A must be 1 so A + A =1 Rules of Boolean Algebra Rule 7: A . A = A ANDing with itself will give the same result Rules of Boolean Algebra Rule 8: A . A = 0 In digital Logic 1 =0 and 0 =1, so AA=0 since one of the inputs must be 0. Rules of Boolean Algebra Rule 9: A = A If you NOT something twice, you are back to the beginning Rules of Boolean Algebra Rule 10: A + AB = A Proof: A + AB = A (1 + B) DISTRIBUTIVE LAW = A∙1 RULE 2: (1+B) = 1 =A RULE 4: A∙1 = A Rules of Boolean Algebra Rule 11: A + AB = A + B If A is 1 the output is 1 , If A is 0 the output is B Proof: A + AB = (A + AB) + AB RULE 10 = (AA + AB) + AB RULE 7 = AA + AB + AA +AB RULE 8 = (A + A)(A + B) FACTORING = 1∙(A + B) RULE 6 =A+B RULE 4 Rules of Boolean Algebra Rule 12: (A + B) (A + C)= A + BC PROOF (A + B)(A +C) = AA + AC +AB +BC DISTRIBUTIVE LAW = A + AC + AB + BC RULE 7 = A(1 + C) +AB + BC FACTORING = A.1 + AB + BC RULE 2 = A(1 + B) + BC FACTORING = A.1 + BC RULE 2 = A + BC RULE 4 Rules of Boolean Algebra SUMMARY - LAWS OF BOOLEAN ALGEBRA [1] COMMUTATIVE: A+B=B+A AB = BA [2] ASSOCIATIVE: A + (B + C) = (A + B) + C A(BC) = (AB)C [3] DISTRIBUTIVE: A(B + C) = AB + AC (A + B)(C + D) = AC + AD + BC + BD Rules of Boolean Algebra SUMMARY - RULES OF BOOLEAN ALGEBRA [1] A + 0 = A [7] A.A = A [2] A + 1 = 1 [8] A.A = 0 [3] A.0 = 0 [9] A = A [4] A.1 = A [10] A + AB = A [5] A + A = A [11] A + AB = A + B [6] A + A = 1 [12] (A + B)(A + C) = A + BC Rules of Boolean Algebra SUMMARY - RULES OF BOOLEAN ALGEBRA [1] A + 0 = A [7] A.A = A [2] A + 1 = 1 [8] A.A = 0 [3] A.0 = 0 [9] A = A [4] A.1 = A [10] A + AB = A [5] A + A = A [11] A + AB = A + B [6] A + A = 1 [12] (A + B)(A + C) = A + BC DeMorgan’s Theorems Two most important theorems of Boolean Algebra were contributed by De Morgan Extremely useful in simplifying expression in which product or sum (POS) of variables is inverted The TWO theorems are: X.Y = X + Y & X+Y = X . Y DeMorgan’s Theorems X.Y = X + Y (a) & (b) Equivalent circuit implied by the theorem (c) Alternative symbol for the NAND function (d) Truth table that illustrates DeMorgan’s Theorem Input (d) Output X Y XY X+Y 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 DeMorgan’s Theorems X+Y = X . Y (a) & (b) Equivalent circuit implied by the theorem (c) Alternative symbol for the NOR function (d) Truth table that illustrates DeMorgan’s Theorem Input (d) Output X Y X+Y XY 0 0 1 1 0 1 0 0 1 0 0 0 1 1 0 0