231 Lecture 4

what is demorgan's theorem

(X . Y)' = X' + Y' ----------- X + X'.Y = X + Y (X + Y)' = X' . Y' ------------ X . (X' + Y) = X . Y

• Boolean (or switching) algebra can be used to: (3)

1. Simplify complex Boolean expressions. • Leads to the design of less costly digital logic circuits. 2. Manipulate (or change the form of) Boolean expressions. • Leads to the design of different logic circuits that perform the same functionality (or behavior). 3. Prove the equivalency of two Boolean expressions. • Provides a method by which two digital logic circuits can be shown to perform the same functionality (or behavior)

Use Boolean algebra to simplify the given Boolean expression. • Determine the minimum (i.e. simplest) expression. • Hint: use the Combining property. F(A,B,C) = ABC + A'BC a) F = BC b) F = AC c) F = A'C d) F = AB e) F = A'B

A

Which of the following is a maxterm of the function F(A,B,C,D)? a) A + B + C + D b) A' + C' c) B + C + D' d) C + D e) A' + B' + D

A

VHDL descriptions include: (2)

An entity to specify the inputs and outputs of the circuit (or system). An architecture to describe the function of the circuit (or system).

VHDL can describe circuits at which 2 different levels of abstraction

Behavioral (dataflow and algorithmic) Structural

Equivalency can be proven using 2 things

Boolean algebra truth table

Showed that Boolean algebra provides an effective means of describing circuits built from switches (and logic gates). Switching Algebra (a 2-valued algebra).

Claude Shannon

Which of the following is a minterm of the function F(A,B,C,D)? a) A'B b) AB'C c) A'C'D' d) ABC'D e) B'D

D

what is first step in designing a minimal circuit

Deriving product of maxterms expression i

what is first step in designing a minimal circuit

Deriving sum of minterms expression

Objective of logic circuit designer

Design a minimum-cost circuit that meets the logical and timing specs. To do so, its essential that the Boolean expression be minimized first.

Use Boolean algebra to simplify the given Boolean expression. • Determine the minimum (i.e. simplest) expression. • Hint: use the Distributive property. F(A,B,C,D) = (A + B)(C + D) a) F = ABCD b) F = AB + CD c) F = AC + BD d) F = A + BC + D e) F = AC + AD + BC + BD

E (use FOIL)

• Developed 10 postulates of Boolean algebra (axioms) which are assumed to be true. • All theorems of switching algebra can be derived from these postulates.

Edward Huntington

symbol for minterms -- how to come up with sum of minterms

F = Σ mi , where f i = 1 example would be:F(A,B,C) = Σ m(1, 2, 4)

Published a method for the algebraic description of processes involved in logical thought and reasoning. Boolean Algebra (an n-valued algebra).

George Boole

what is a circuit diagram

Graphical representation of a logic function. • Shows inputs, output(s), and logic operators. • Complex logic functions realized using a network of logic gates.

a logic cirucit includes 4 elements:

Inputs Outputs Functional description Timing specification

Switching Functions • aka.

Logic Functions

what is a Boolean Expression

Mathematical description of a switching function. • Output(s) specified in terms of the inputs. • Composed of variables (inputs) and operators (gates). • Evaluated by substituting 0 or 1 for each variable. • Truth table represents the evaluation of the Boolean expression for all combinations of the input variables

F(A,B,C,D) = (A'+B+D)(A+C')(B+C'+D')

Product of Maxterms - POS

All logic functions can be described using product of maxterms. where

Product of maxterms is unique. • However, it may not be minimal

what is a timing diagram

Representation in the time domain. • Shows inputs and outputs as a function of time. • Each signal represents one input or output

F(A,B,C,D) = ABC' + A'CD + B'D'

Sum of Minterms - SOP

In general, Boolean expressions can be written in one of two standard forms:

Sum of Minterms -- Sum of Products (SOP) Product of Maxterms - Product of Sums (POS)

All logic functions can be described using sum of minterms where

Sum of minterms is unique. • However, it may not be minimal.

maxterms aka

The canonical product of sums.

minterms AKA

The canonical sum of products.

A switching function can be described by the following:

Truth table • Boolean expression • Circuit diagram • Timing diagram • VHDL • Each can be used to derive the others.

Problems with Boolean Algebra

Useful for manual simplification of simple Boolean expressions. However, it lacks a well-defined process (or set of steps) Furthermore, there is no easy way to determine when the given Boolean expression has been simplified to A minimum number of terms • And a minimum number of literals (in each term).

what is absorption property

X + (X . Y) = X X . (X + Y) = X

what are the following single variable theorems of boolean algrbra X + 1 = X + 0 = X + X = X + X' = X'' =

X + 1 = 1 X + 0 = X X + X = X X + X' = 1 X'' = X

what is distributive property

X . (Y + Z) = X.Y + X.Z X + (Y . Z) = (X+Y).(X+Z)

what is associative property

X . (Y . Z) = (X . Y) . Z

what are the following the single-variable theorems of Boolean algebra X . 0 = X . 1 = X . X = X . X' = X'' =

X . 0 = 0 X . 1 = X X . X = X X . X' = 0 X'' = X

what is commutative property

X . Y = Y . X X + Y = Y + X

what is combining theorem

X.Y + X.Y' = X (X + Y).(X + Y') = X

Which of the following statements about a Truth Table is true ? a) It is the tabular representation of a logic function. b) It is the mathematical description of a logic function. c) It is the graphical representation of a logic function. d) It is the representation of a logic function in the time domain. e) It is the representation of a logic function using a formal language.

a

For which input combination does the given minterm for function F(A,B,C,D) evaluate to 1? Minterm = AB'C'D a) ABCD = 1111 b) ABCD = 1100 c) ABCD = 1001 d) ABCD = 0101 e) ABCD = 0110 f) Cannot be determined.

c

Use Boolean algebra to simplify the given Boolean expression. • Determine the minimum (i.e. simplest) expression. • Hint: use the Distributive property. F(A,B,C) = (A' + B)(A + C) a) F = A'C + AB b) F = A'B + BC c) F = AB + BC + A'C d) F = A'A + A'C + AB + BC e) Cannot be simplified.

c

Which of the following is NOT a minterm of the function F(A,B,C,D)? a) ABCD b) A'B'C'D' c) B'CD' d) A'BC'D e) ABC'D'

c

• Use Boolean algebra to simplify the given Boolean expression. • Determine the minimum (i.e. simplest) expression. • Hint: use DeMorgan's Theorems. F(A,B,C) = (AB' + C)' a) F = A' + BC' b) F = AC + B'C c) F = A'C' + BC' d) F = A'B + C' e) F = (AB')' + C'

c

Minterms and Maxterms for function F are

complementary

Use Boolean algebra to simplify the given Boolean expression. • Determine the minimum (i.e. simplest) expression. • Hint: use DeMorgan's Theorems. F(A,B,C) = ( (A + B')(CD) )' a) F = A'B + C'D' b) F = AB' + (C' + D') c) F = ACD' + B'CD' d) F = A'B + C' + D' e) F = A'C' + BC' + A'D' + BD'

d

Use Boolean algebra to simplify the given Boolean expression. • Determine the minimum (i.e. simplest) expression. • Hint: use the Combining property F(A,B,C) = ABC + AB'C + A'BC + ABC' a) F = AC + A'BC + ABC' b) F = AB'C + BC + ABC' c) F = A'BC + AB'C + AB d) F = AB + AC + BC e) F = AB'C + A'BC + ABC'

d

Which of the following is NOT a maxterm of the function F(A,B,C,D)? a) A + B' + C' + D' b) A' + B' + C + D c) A + B + C + D d) C + D e) A' + B + C + D'

d

Use Boolean algebra to simplify the given Boolean expression. • Determine the minimum (i.e. simplest) expression. • Hint: use the Combining property. F(A,B,C) = AB'C + ABC' a) F = AC b) F = BC' c) F = AB d) F = B'C e) Cannot be simplified.

e

• Use Boolean algebra to simplify the given Boolean expression. • Determine the minimum (i.e. simplest) expression. • Hint: use the Combining property. F(A,B,C) = AB'C + A'BC a) F = AB' b) F = B'C c) F = AC d) F = C e) Cannot be simplified.

e

• Which maxterm of function F(A,B,C,D) evaluates to 0 for the given input combination? ABCD = 0101 a) A' + B + C' + D b) A + B + C + D c) A + B + C' + D' d) A' + B' + C' + D' e) A + B' + C + D' f) Cannot be determined.

e

explain maxterms

focus on 0's put inverter on 1's in truth table

Explain Minterms

focuses on "1" so put complement on 0's in truth table

• Two logic circuits are functionally equivalent if,

for all combinations of the inputs, their outputs are the same.

what is VHDL

formal language for describing the behavior and structure of digital circuits and systems. hardware description language. It describes, and leads to the synthesis of, hardware. It is used to described combinational and sequential logic circuits.

a better tool for manual simplification of Boolean expressions.

k-map

F(A,B,C) = (A+B+C)(A+B'+C')(A'+B+C')

product of maxterms (POS)

Equivalent circuits are described by the

same truth table

what is a truth table

tabular representation of a switching function. • Output of function defined for all combinations of the inputs. • Each row corresponds to one input combination. • For an n-variable function, there are 2n rows.

Deriving the product of maxterms: (symbol)

• Include maxterms for which F = 0. • AND maxterms together. F = Π Mi , ex:F(A,B,C) = Π M(0, 3, 5)