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# boolean algebra

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### boolean algebra

1. 3. <ul><li>WELL LET’S GO ON NOW TO THE MAIN DISH OF OUR LESSON FOR TODAY… </li></ul><ul><li>AND THIS IS ALL ABOUT…. </li></ul>
3. 6. <ul><ul><li>In 1854,George Boole invented two-state algebra, known today as BOOLEAN ALGEBRA. </li></ul></ul><ul><ul><li>Every variable in Boolean Algebra can have only have either of two values: TRUE or FALSE. </li></ul></ul><ul><ul><li>This Algebra had no practical use until Claude Shannon applied it to telephone switching circuits. </li></ul></ul>
4. 7. <ul><ul><li>BOOLEAN ALGEBRA is a branch of mathematics that is directly applicable to digital designs. </li></ul></ul><ul><ul><li>It is a set of elements, a set of operators that act on these elements, and a set of axioms or postulates that govern the actions of these operators on these elements. </li></ul></ul>
5. 8. <ul><li>The Postulates commonly used to define Algebraic Structures are: </li></ul>
6. 9. <ul><li>A set S is closed with respect to a binary operator * if, application of the operator on every pair of elements of S results is an element of S. </li></ul>
7. 10. <ul><li>A binary operator * on a set S is said to be associative when (x * y) * z = x * (y * z) </li></ul><ul><li>For all x ,y ,z , that are elements of the set S . </li></ul>
8. 11. <ul><li>A binary operator * on a set S is said to be commutative when x * y = y * x , </li></ul><ul><li>For all x, y, that are elements of the set S . </li></ul>
9. 12. <ul><li>A set S is said to be Identity element with respect a binary operator * on S if there exist an element e in the set S such that x * e = x </li></ul><ul><li>For all x that are elements of the set S. </li></ul>
10. 13. <ul><li>It is said to have an inverse elements when for every x that is an element of the set S , there exists an element x’ that is a member of the set S such that </li></ul><ul><li> x * x = e. </li></ul>
11. 14. <ul><li>If * and “ .” Are two binary operators on a set S, * is said to be distributive over “ .” when x * (y * z) = (x * y) . (x * z) </li></ul><ul><li>For all x, y, z, that are elements of the set S . </li></ul>
12. 15. <ul><li>BASIC THEOREMS OF BOOLEAN ALGEBRA </li></ul>
13. 18. Solution Reason x + x = ( x + x ) 1 Basic identity (b) = ( x + x ) ( x + x’) Basic identity (a) = x + xx’ Distributive (b) = x + 0 Basic identity (b) = x Basic identity (a)
14. 19. Solution Reason x . x = xx + 0 Basic identity (a) = xx + xx’ Basic identity (b) = x (x + x’ ) Distributive (a) = x . 1 Basic identity (a) = x Basic identity (b)
15. 20. Solution Reason x + 1 = (x+1) . 1 Basic identity (b) = (x+1) . (x + x’) Basic identity (a) = x + 1x’ Distributive (b) = x +x’ Basic identity (b) = 1 Basic identity (a)
16. 21. Solution Reason x . 0 = x.0 + 0 Basic identity (a) = x.0 + xx’ Basic identity (b) = x ( 0 + x’ ) Distributive (a) = x ( x’ ) Basic identity (a) = 0 Basic identity (b)
17. 22. <ul><li>From postulate 5, we have x + x’ = 1 and x . X’ = 0, which defines the complement of x. The complement of x’ is x and is also ( x’ )’. </li></ul><ul><li>Therefore, we have that ( x’)’ = x. </li></ul>
18. 23. Solution Reason x + xy = x.1 + xy Basic identity (a) = x( 1 + y ) Distributive (a) = x (y+ 1) Commutative (a) = x .1 Basic identity (a) = x Basic identity (b)
19. 24. Solution Reason x ( x + y ) = xx + xy Distributive (a) = x + xy Basic identity (b) = x1 + xy Basic identity (b) = x ( 1 + y ) Distributive (a) = x.1 Basic identity (a) = x Basic identity (b)
20. 26. <ul><ul><li>To complement a variable is to reverse its value. </li></ul></ul><ul><ul><li>Thus , </li></ul></ul><ul><ul><li>if x=1, then, x’=0 </li></ul></ul><ul><ul><li> if x=0, then, x’=1 </li></ul></ul>
21. 27. <ul><ul><ul><li>This operation is equivalent to a logical OR operation. The (+) plus symbol is used to indicate addition or Oring. </li></ul></ul></ul><ul><ul><ul><li>0 + 0 = 0 </li></ul></ul></ul><ul><ul><ul><li>0 + 1 = 1 </li></ul></ul></ul><ul><ul><ul><li>1 + 1 = 1 </li></ul></ul></ul>
22. 28. <ul><ul><li>Is equivalent to a logical AND operation. </li></ul></ul><ul><ul><li>0 . 0 = 0 </li></ul></ul><ul><ul><li>0 . 1 = 0 </li></ul></ul><ul><ul><li>1 . 1 = 1 </li></ul></ul>
23. 29. <ul><li>The theorems of BOOLEAN ALGEBRA can be shown to hold true by means of a truth table. </li></ul><ul><li>If a function has N inputs, there are 2 raise to N possible combinations of these inputs and there will be 2 raise to N entries in the truth table. </li></ul>
24. 30. <ul><ul><li>Solution: </li></ul></ul><ul><ul><li>In this example you have a value of N equal to 2, </li></ul></ul><ul><ul><li>Therefore the possible combinations if 2 raise to N </li></ul></ul><ul><ul><li>is 4. Let x and y represent the variables. </li></ul></ul>x y x+ y x( x+y ) 0 0 0 0 0 1 1 0 1 0 1 1 1 1 1 1
25. 31. <ul><li>HAKUNAH MATATAH! </li></ul><ul><li>-  - </li></ul><ul><li> -Rizan ‘2012 </li></ul>