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# Ee2 chapter7 boolean_algebra

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### Ee2 chapter7 boolean_algebra

1. 1. IT2001PAEngineering Essentials (2/2)Chapter 7 - Boolean Algebra Lecturer Namelecturer_email@ite.edu.sg Nov 20, 2012Contact Number
2. 2. Chapter 7 - Boolean AlgebraLesson ObjectivesUpon completion of this topic, you should be able to: State the rules and functions of Boolean algebra. IT2001PA Engineering Essentials (2/2) 2
3. 3. Chapter 7 - Boolean AlgebraSpecific Objectives Students should be able to :  State the function of Boolean algebra.  State the 9 equalities of Boolean algebra.  State the Commutative Law.  State the Associative Law.  State the Distributive Law. IT2001PA Engineering Essentials (2/2)
4. 4. Chapter 7 - Boolean AlgebraIntroduction Boolean Algebra is a set of algebraic rules, named after mathematician George Boole, in which TRUE and FALSE are equated to ‘0’ and ‘1’. Boolean algebra includes a series of operators (AND, OR, NOT, NAND (NOT AND), NOR, and XOR (exclusive OR)), which can be used to manipulate TRUE and FALSE values. It is the basis of computer logic because the truth values can be directly associated with bits. IT2001PA Engineering Essentials (2/2) 4
5. 5. Chapter 7 - Boolean AlgebraBoolean Algebra Laws Frequently, a Boolean expression is not in its simplest form. Boolean expressions can be simplified, but we need identities or laws that apply to Boolean algebra instead of regular algebra. These identities can be applied to single Boolean variables as well as Boolean expressions. IT2001PA Engineering Essentials (2/2) 5
6. 6. Chapter 7 - Boolean AlgebraCommutative Law A+B=B+A A*B = B*AAssociative Law A+B+C = A+(B+C) = (A+B)+C = B+(A+C) A*B*C = A*(B*C) = (A*B)*C = B*(A*C)Distributive Law A(B+C) = A*B + A*C (A+B)*(C+D) = A*C + A*D + B*C + B*D IT2001PA Engineering Essentials (2/2) 6
7. 7. Chapter 7 - Boolean AlgebraCommutative Law The commutative law allows the change in position (reordering) of an ANDed or ORed variable. IT2001PA Engineering Essentials (2/2) 7
8. 8. Chapter 7 - Boolean AlgebraAssociative Law The Associative Law allows the regrouping of variables. IT2001PA Engineering Essentials (2/2) 8
9. 9. Chapter 7 - Boolean AlgebraDistributive Law The Distributive Law shows how OR distributes over AND and vice versa. IT2001PA Engineering Essentials (2/2) 9
10. 10. Chapter 7 - Boolean AlgebraBoolean Algebra Theorems Boolean theorems can help to simplify logic expression and logic circuits. IT2001PA Engineering Essentials (2/2) 10
11. 11. Chapter 7 - Boolean AlgebraBoolean Algebra Equalities or Identities (a) A . 0 =0 (j) A + AB = A + B (b) A . 1 =A A + AB = A + B (c) A . A =A (d) A . A =0 The variable A may represent (e) A + 0 = A an expression containing more (f) A + 1 = 1 than one variable. For instance, XY(XY) (g) A + A = A (h) A + A = 1 Let A = XY, (i) A =A then XY(XY) = A . A = 0 IT2001PA Engineering Essentials (2/2) 11
12. 12. Chapter 7 - Boolean AlgebraBoolean Algebra Equalities or Identities A*0 = 0 A*1 = A A A X = A*0 = 0 X = A*1 = A 0 1 Anything ANDed with a 0 is Anything ANDed with a 1 is • equal to 0. • equal to itself. A*A = A A*A = 0 A A A X = A*A = A 1 X = A*A = 0 Anything ANDed with itself is Anything ANDed with its own • equal to itself. • complement equals 0. IT2001PA Engineering Essentials (2/2) 12
13. 13. Chapter 7 - Boolean AlgebraBoolean Algebra Equalities or Identities A+0 = A A+1 = 1 A A X = A+0 = A X = A+1 = 1 0 1 Anything ORed with a 0 is Anything ORed with a 1 is • equal to itself. • equal to 1. A+A = A A+A = 0 A A A X = A+A = A 1 X = A+A = 1 Anything ORed with itself is Anything ORed with its own • equal to itself. • complement equals 1. IT2001PA Engineering Essentials (2/2) 13
14. 14. Chapter 7 - Boolean AlgebraBoolean Algebra Equalities or Identities A=A A A X=A=A A variable that is complemented twice will • return to its original IT2001PA Engineering Essentials (2/2) 14
15. 15. Chapter 7 - Boolean AlgebraElimination Law Equivalence is demonstrated by showing the truth table derived from the expression on the left side of the equation matches that on the right side. IT2001PA Engineering Essentials (2/2) 15
16. 16. Chapter 7 - Boolean AlgebraNext Lesson IT2001PA Engineering Essentials (2/2) 16