Topic1: Boolean Algebra José Nelson Amaral CMPUT329 - Fall 2003 Topic1: Boolean Algebra José Nelson Amaral CMPUT 329 Computer Organization and Architecture II

CMPUT 329 Computer Organization and Architecture II Reading Assignment Wakerly, Chapter 4 (Sections 4.1.1, 4.1.2, 4.1.3) CMPUT 329 Computer Organization and Architecture II

What is a switching network? X1 Xm X2 Z1 Zm Z2 Combinatorial Network: A stateless network. The output is completely determined by the values of the input. Sequential Network: The network stores an internal state. The output is determined by the input, and by the internal state. CMPUT 329 Computer Organization and Architecture II

Logic Functions: Boolean Algebra INVERTER X X’ If X=0 then X’=1 If X=1 then X’=0 A B C=A·B If A=1 AND B=1 then C=1 otherwise C=0 AND OR A B C=A+B If A=1 OR B=1 then C=1 otherwise C=0 CMPUT 329 Computer Organization and Architecture II

Boolean expressions and logic circuits Any Boolean expression can be implemented as a logic circuit. X = [A(C+D)]’+BE C D C+D A A(C+D) [A(C+D)]’ [A(C+D)]’+BE B E BE CMPUT 329 Computer Organization and Architecture II

Basic Theorems: Operations with 0 and 1 X+0 = X X C=X X+1 = 1 X 1 C=1 X 1 C=X X·1 = X X C=0 X·0 = 0 CMPUT 329 Computer Organization and Architecture II

Basic Theorems: Idempotent Laws X+X = X X C=X X C=X X·X = X CMPUT 329 Computer Organization and Architecture II

Basic Theorems: Involution Law (X’)’=X B X C=X CMPUT 329 Computer Organization and Architecture II

Basic Theorems: Laws of Complementarity X+X’ = 1 X X’ C=1 X X’ C=0 X·X’ = 0 CMPUT 329 Computer Organization and Architecture II

Expression Simplification using the Basic Theorems X can be an arbitrarily complex expression. Simplify the following boolean expressions as much as you can using the basic theorems. (AB’ + D)E + 1 = 1 (AB’ + D)(AB’ + D)’ = 0 (AB + CD) + (CD + A) + (AB + CD)’ = 1 (AB’ + D)E + 1 = (AB’ + D)(AB’ + D)’ = (AB + CD) + (CD + A) + (AB + CD)’ = CMPUT 329 Computer Organization and Architecture II

CMPUT 329 Computer Organization and Architecture II Associative Law (X+Y)+Z = X+(Y+Z) X Y Z C CMPUT 329 Computer Organization and Architecture II

CMPUT 329 Computer Organization and Architecture II Associative Law (XY)Z = X(YZ) X Y Z C Y Z X C CMPUT 329 Computer Organization and Architecture II

First Distributive Law X(Y+Z) = XY+XZ CMPUT 329 Computer Organization and Architecture II

First Distributive Law X(Y+Z) = XY+XZ CMPUT 329 Computer Organization and Architecture II

First Distributive Law X(Y+Z) = XY+XZ CMPUT 329 Computer Organization and Architecture II

First Distributive Law X(Y+Z) = XY+XZ CMPUT 329 Computer Organization and Architecture II

First Distributive Law X(Y+Z) = XY+XZ CMPUT 329 Computer Organization and Architecture II

Second Distributive Law X+YZ = (X+Y)(X+Z) CMPUT 329 Computer Organization and Architecture II

Second Distributive Law X+YZ = (X+Y)(X+Z) CMPUT 329 Computer Organization and Architecture II

Second Distributive Law (A different proof) (X + Y)(X + Z) = X(X + Z) + Y(X + Z) (using the first distributive law) = XX + XZ + YX + YZ (using the first distributive law) = X + XZ + YX + YZ (using the idempotent law) = X·1 + XZ + YX + YZ (using the operation with 1 law) = X(1 + Z + Y) + YZ (using the first distributive law) = X·1 + YZ (using the operation with 1 law) = X + YZ (using the operation with 1 law) CMPUT 329 Computer Organization and Architecture II

Simplification Theorems XY + XY’ = X XY + XY’ = X(Y + Y’) = X·1 = X (X + Y)(X + Y’) = X (X + Y)(X + Y’) = XX + XY’ + YX + YY’ = X + X(Y’ + Y) + 0 = X + X·1 = X X + XY = X X(1 + Y) = X·1 = X X(X + Y) = X X(X + Y) = XX + XY = X·1 + XY = X(1 + Y) = X·1 = X XY’ + Y = X + Y (using the second distributive law) XY’ + Y = Y + XY’ = (Y + X)(Y + Y’) = (Y + X)·1 = X + Y (X + Y’)Y = XY XY + Y’Y = XY + 0 = XY CMPUT 329 Computer Organization and Architecture II

Examples Simplify the following expressions: W = [M + N’P + (R + ST)’][M + N’P + R + ST] X = M + N’P Y = R + ST W = (X + Y’)(X + Y) W = XX + XY + Y’X + Y’Y W = X·1 + XY + XY’ + 0 W = X + X(Y + Y’) = X + X·1 = X W = M + N’P CMPUT 329 Computer Organization and Architecture II