1. © 2003-2008 BYU 03 BA1 Page 1 ECEn 224 Boolean Algebra – Part 1

2. © 2003-2008 BYU 03 BA1 Page 2 ECEn 224 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems Apply Boolean Algebra to: –Simplifying Expressions –Multiplying Out Expressions –Factoring Expressions

3. © 2003-2008 BYU 03 BA1 Page 3 ECEn 224 A New Kind of Algebra

4. © 2003-2008 BYU 03 BA1 Page 4 ECEn 224 Truth Tables A truth table provides a complete enumeration of the inputs and the corresponding output for a function. If there are n inputs, There will be 2 n rows In the table. Unlike with regular algebra, full enumeration is possible (and useful) in Boolean Algebra

5. © 2003-2008 BYU 03 BA1 Page 5 ECEn 224 Complement Operation Also known as invert or not. The inverse of x is written as x’ or x.

6. © 2003-2008 BYU 03 BA1 Page 6 ECEn 224 Logical AND Operation “” denotes AND Output is true iff all inputs are true

7. © 2003-2008 BYU 03 BA1 Page 7 ECEn 224 Logical OR Operation “+” denotes OR Output is true if any inputs are true

8. © 2003-2008 BYU 03 BA1 Page 8 ECEn 224 Boolean Expressions Boolean expressions are made up of variables and constants combined by AND, OR and NOT Examples: 1 A ’ A B C+D AB A(B+C) AB’+C A B is the same as AB ( is omitted when obvious ). Parentheses are used like in regular algebra for grouping. NOT has highest precedence, followed by AND then OR. A term is a grouping of variables ANDed together. A literal is each instance of a variable or constant. This expression has 4 variables, 3 terms, and 6 literals: a’b + bcd + a

9. © 2003-2008 BYU 03 BA1 Page 9 ECEn 224 Boolean Expressions Each Boolean expression can be specified by a truth table which lists all possible combinations of the values of all variables in the expression. F = A’ + B C

10. © 2003-2008 BYU 03 BA1 Page 10 ECEn 224 Boolean Expressions From Truth Tables Each 1 in the output of a truth table specifies one term in the corresponding boolean expression. The expression can be read off by inspection… F is true when: A is false AND B is true AND C is false OR A is true AND B is true AND C is true F = A’BC’ + ABC

11. © 2003-2008 BYU 03 BA1 Page 11 ECEn 224 Another Example F = A’B’C + A’BC’ + AB’C’ + ABC F = ?

12. © 2003-2008 BYU 03 BA1 Page 12 ECEn 224 Yet Another Example F = A’B’ + A’B + AB’ + AB = 1 (F is always true) There may be multiple expressions for any given truth table. F = ?

13. © 2003-2008 BYU 03 BA1 Page 13 ECEn 224 Converting Boolean Functions to Truth Tables F = AB + BC AB BC

14. © 2003-2008 BYU 03 BA1 Page 14 ECEn 224 Converting Boolean Functions to Truth Tables F = AB + BC AB BC

15. © 2003-2008 BYU 03 BA1 Page 15 ECEn 224 Some Basic Boolean Theorems From these… We can derive these…

16. © 2003-2008 BYU 03 BA1 Page 16 ECEn 224 Proof Using Truth Tables Truth Tables can be used to prove that 2 Boolean expressions are equal. If the two expressions have the same value for all possible combinations of input variables, they are equal. X Y’ + Y = X + Y = These two expressions are equal.

17. © 2003-2008 BYU 03 BA1 Page 17 ECEn 224 Basic Boolean Algebra Theorems Here are the first 5 Boolean Algebra theorems we will study and use.

18. © 2003-2008 BYU 03 BA1 Page 18 ECEn 224 Basic Boolean Algebra Theorems While these laws don’t seem very exciting, they can be very useful in simplifying Boolean expressions: Simplify: (M N ’ + M ’ N) P + P ’ + 1 X+ 1 1

19. © 2003-2008 BYU 03 BA1 Page 19 ECEn 224 Commutative Laws X Y = Y X X + Y = Y + X Associative Laws (X Y) Z = X (Y Z) = X Y Z (X + Y) + Z = X + (Y + Z ) = X + Y + Z Just like regular algebra

20. © 2003-2008 BYU 03 BA1 Page 20 ECEn 224 Distributive Law X ( Y + Z ) = X Y + X Z Prove with a truth table: = Just like regular algebra

21. © 2003-2008 BYU 03 BA1 Page 21 ECEn 224 Other Distributive Law X + Y Z = ( X + Y ) ( X + Z ) NOT like regular algebra! Proof: ( X + Y ) ( X + Z )= X ( X + Z) + Y ( X + Z ) = X X + X Z + Y X + Y Z = X + X Z + X Y + Y Z = X 1 + X Z + X Y + Y Z = X ( 1 + Z + Y ) + Y Z = X 1 + Y Z = X + Y Z

22. © 2003-2008 BYU 03 BA1 Page 22 ECEn 224 Simplification Theorems X Y + X Y’ = X( X + Y ) (X + Y’) = X X + X Y = XX ( X + Y ) = X X + Y X’ = X + YX ( X’ + Y ) = X Y These are useful for simplifying Boolean Expressions. The trick is to find X and Y. ( A’ + B + CD) (B’ + A’ + CD) ( A’ + CD + B) (A’ + CD + B’) A’ + CD Rearrange terms (X + Y)(X + Y’) = X

23. © 2003-2008 BYU 03 BA1 Page 23 ECEn 224 Multiplying Out All terms are products only (no parentheses) A B C ’ + D E + F G H Yes A B + C D + E Yes A B + C ( D + E ) No Multiplied out = sum-of-products form (SOP)

24. © 2003-2008 BYU 03 BA1 Page 24 ECEn 224 Multiplying Out - Example ( A’ + B )( A’ + C )( C + D ) Use ( X + Y )( X + Z ) = X + Y Z ( A’ + B C )( C + D ) Multiply Out A’ C + A’ D + B C C + B C D Use X X = X A’ C + A’ D + B C + B C D Use X + X Y = X A’ C + A’ D + BC Using the theorems may be simpler than brute force. But, brute force does work…

25. © 2003-2008 BYU 03 BA1 Page 25 ECEn 224 Factoring Final form is products only All sum terms are single variables only ( A + B + C ’ ) ( D + E ) Yes ( A + B )( C + D ’ )E ’ F Yes ( A + B + C ’ ) ( D + E ) + H No ( A ’ + BC ) ( D + E ) No Factored out = product-of-sums form (POS)

26. © 2003-2008 BYU 03 BA1 Page 26 ECEn 224 Factoring - Example A B + C D Use X + Y Z = ( X + Y )( X + Z ) ( A B + C )( A B + D ) Use X + Y Z = ( X + Y )( X + Z ) again ( A + C )( B + C )( A B + D ) And again ( A + C )( B + C )( A + D )( B + D ) Sum of Products (SOP) Product of Sums (POS)

27. © 2003-2008 BYU 03 BA1 Page 27 ECEn 224 POS vs. SOP Any expression can be written either way Can convert from one to another using theorems Sometimes SOP looks simpler –AB + CD = ( A + C )( B + C )( A + D )( B + D ) Sometimes POS looks simpler –(A + B)(C + D) = BD + AD + BC + AC SOP will be most commonly used in this class but we must learn both

28. © 2003-2008 BYU 03 BA1 Page 28 ECEn 224 Duality If an equality is true its dual will be true as well To form the dual: –AND  OR –Invert constant 0’s or 1’s –Do NOT invert variables Because these are true These are also true

29. © 2003-2008 BYU 03 BA1 Page 29 ECEn 224 Theorem Summary X + 0 = XX 1 = X X + 1 = 1X 0 = 0 X + X = XX X = X X + X’ = 1X X’ = 0 (X’)’ = X XY + XY’ = X(X + Y) (X + Y’) = X X + XY = XX(X + Y) = X X + YX’ = X + YX(Y + X’) = X Y XY + X’Z + YZ = XY + X’Z(X + Y)(X’ + Z)(Y + Z) = (X + Y)(X’ + Z) For each theorem on the left, the dual is listed on the right. HINT: when forming duals, first apply parentheses around all AND terms.