Boolean Algebra – Part 1 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 ECEn 224

A New Kind of Algebra 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 2n rows In the table. Unlike with regular algebra, full enumeration is possible (and useful) in Boolean Algebra ECEn 224

Complement Operation Also known as invert or not. The inverse of x is written as x’ or x. ECEn 224

Logical AND Operation “•” denotes AND Output is true iff all inputs are true ECEn 224

Logical OR Operation “+” denotes OR Output is true if any inputs are true 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 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 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 ECEn 224

Another Example F = ? F = A’B’C + A’BC’ + AB’C’ + ABC ECEn 224

There may be multiple expressions for any given truth table. Yet Another Example F = ? F = A’B’ + A’B + AB’ + AB = 1 (F is always true) There may be multiple expressions for any given truth table. ECEn 224

Converting Boolean Functions to Truth Tables F = AB + BC BC AB BC ECEn 224

Converting Boolean Functions to Truth Tables F = AB + BC BC AB BC ECEn 224

Some Basic Boolean Theorems From these… We can derive these… 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. ECEn 224

Basic Boolean Algebra Theorems Here are the first 5 Boolean Algebra theorems we will study and use. 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 ECEn 224

Commutative Laws Associative 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 ECEn 224

Just like regular algebra Distributive Law X ( Y + Z ) = X Y + X Z Prove with a truth table: = Just like regular algebra ECEn 224

Other Distributive Law X + Y Z = ( X + Y ) ( X + Z ) ( 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 Proof: NOT like regular algebra! ECEn 224

Simplification Theorems X Y + X Y’ = X ( X + Y ) (X + Y’) = X X + X Y = X X ( X + Y ) = X X + Y X’ = X + Y 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’) Rearrange terms A’ + CD (X + Y)(X + Y’) = X ECEn 224

Multiplied out = sum-of-products form (SOP) 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) 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… 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) ECEn 224

Factoring - Example A B + C D ( A B + C )( A B + D ) Sum of Products (SOP) 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 ) Product of Sums (POS) 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 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 ECEn 224

For each theorem on the left, the dual is listed on the right. 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. ECEn 224