Boolean Algebra Part 1

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1 Boolean Algebra Part 1 Page 1

2 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 Page 2

3 A New Kind of Algebra Regular Algebra Boolean Algebra Values Integers Real Numbers Complex Numbers Operators + - / Logarithm, etc. Zero (0) One (1) AND OR Complement Page 3

4 Truth Tables A truth table provides a complete enumeration of the inputs and the corresponding output for a function. A B F 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 Page 4

5 Complement Operation Also known as invert or not. The inverse of x is written as x or x. x x' Page 5

6 Logical AND Operation denotes AND Output is true iff all inputs are true A B Q=A B Page 6

7 Logical OR Operation + denotes OR Output is true if any inputs are true A B Q=A+B Page 7

8 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 Page 8

9 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 A B C A' B C F = A' + B C Page 9

10 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 A B C F 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 Page 10

11 Another Example A B C F F =? F = A B C + A BC + AB C + ABC Page 11

12 Yet Another Example A B F F =? F = A B + A B + AB + AB = 1 (F is always true) There may be multiple expressions for any given truth table. Page 12

13 Converting Boolean Functions to Truth Tables F = AB + BC BC BC A B C F AB Page 13

14 Converting Boolean Functions to Truth Tables F = AB + BC BC BC A B C F AB Page 14

15 Some Basic Boolean Theorems A 0 A = 0 A B Q = A B A B Q = A + B From these We can derive these A 1 A = A A 0 + A = A A 1 + A = Page 15

16 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 X Y Y' X Y' X Y' + Y X + Y These two expressions are equal. = Page 16

17 Basic Boolean Algebra Theorems Here are the first 5 Boolean Algebra theorems we will study and use. X + 0 = X X 1 = X X + 1 = 1 X 0 = 0 X + X = X X X = X X + X' = 1 X X' = 0 ( X' )' = X Page 17

18 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 Page 18

19 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 Page 19

20 Distributive Law X ( Y + Z ) = X Y + X Z Prove with a truth table: X Y Z Y + Z X ( Y + Z ) X Y X Z X Y + X Z Just like regular algebra = Page 20

21 Other Distributive Law X + Y Z = ( X + Y ) ( X + Z ) 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 NOT like regular algebra! Page 21

22 Simplification Theorems X Y + X Y = X X + X Y = X X + Y X = X + Y ( X + Y ) (X + Y ) = X X ( X + Y ) = X X ( 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 Page 22

23 Multiplying Out All terms are products only (no parentheses) A B C + D E + F G H A B + C D + E A B + C ( D + E ) Yes Yes No Multiplied out = sum-of-products form (SOP) Page 23

24 Multiplying Out - Example ( A + B )( A + C )( C + D ) ( A + B C )( C + D ) A C + A D + B C C + B C D Use ( X + Y )( X + Z ) = X + Y Z Multiply Out Use X X = X A C + A D + B C + B C D A C + A D + BC Use X + X Y = X Using the theorems may be simpler than brute force. But, brute force does work Page 24

25 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) Page 25

26 Factoring - Example A B + C D ( A B + C )( A B + D ) Sum of Products (SOP) ( A + C )( B + C )( A B + D ) Use X + Y Z = ( X + Y )( X + Z ) Use X + Y Z = ( X + Y )( X + Z ) again ( A + C )( B + C )( A + D )( B + D ) And again Product of Sums (POS) Page 26

27 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 Page 27

28 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 X + 0 = X X 1 = X X + 1 = 1 X 0 = 0 X + X = X X X = X X + X' = 1 X X' = 0 Page 28

29 Theorem Summary For each theorem on the left, the dual is listed on the right. X + 0 = X X 1 = X X + 1 = 1 X 0 = 0 X + X = X X X = X X + X = 1 X X = 0 (X ) = X XY + XY = X (X + Y) (X + Y ) = X X + XY = X X(X + Y) = X X + YX = X + Y X(Y + X ) = X Y XY + X Z + YZ = XY + X Z (X + Y)(X + Z)(Y + Z) = (X + Y)(X + Z) HINT: when forming duals, first apply parentheses around all AND terms. Page 29

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