DDM Chapter 5

True.

(T/F) De Morgan's law can be applied repeatedly to eliminate all the addition operations from any Boolean expression.

True

(T/F) The complement operation is applied as soon as the entire expression under the bar is evaluated.

true

(T/F) x • *Bar on top of y+z* y + z is evaluated as x • *Bar on top of y+z* (y + z)

Boolean satisfiability

(called SAT for short) takes a Boolean expression as input and asks whether it is possible to set the values of the variables so that the expression evaluates to 1.

xy

(x ↑ y)' = _______

1

1 + 0 = ?

1

1 + 1 = ?

0

1 • 0 = ?

1

1 • 1 = ?

0

1' = ?

B, A, A.

1) Which identity is true? A) a = a ↓ a B) a' = a ↓ a 2) Which identity is true? A) x + y = (x ↓ y)' B) x + y = x' ↓ y' 3) Now express x + y with only NOR operations, by using the identity in question 2 applied to x ↓ y. A) x + y = (x ↓ y) ↓ (x ↓ y) B) x + y = (x ↓ x) ↓ (y ↓ y)

conjunctive normal form

A Boolean expression that is a product of sums of literals (abbreviated as CNF).

disjunctive normal form

A Boolean expression that is a sum of products of literals.

b

A Boolean function with three variables is defined in the table below: x y z f(x, y, z) 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 Is xy a minterm for the function? A) Yes B) No

b

A Boolean function with three variables is defined in the table below: x y z f(x, y, z) 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 The Boolean expression for the function specified in the table is a sum of minterms. The minterms for which rows are included in sum? A) 1, 4, 7 B) 1, 4, 5, 7 C) 1, 4, 5, 6, 7

a

A Boolean function with three variables is defined in the table below: x y z f(x, y, z) 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 What is the minterm corresponding to the 7th row in the table? (' denotes complement) A) xyz' B) x'yz' C) y'z

B

A Boolean function with three variables is defined in the table below: x y z f(x, y, z) 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 Which Boolean expression correctly specifies the function in the table? A) x' y' z' + xy' z' + xyz' B) x' y' z' + x'yz + xy' z' + xyz' C) x' y' z' + x'yz + xy' z' + xyz

d1 * d2 * .... * dm

A conjunctive normal form (CNF) expression has the form: _________.

c1 + c2 + .... + cm

A disjunctive normal form (DNF) expression has the form: ________.

A, B, A.

Are the following functions satisfiable? Try finding variables values that cause f = 1, or use logic to conclude f can never be 1. 1) f(x, y, z) = xy + z A) Yes B) No 2) f(x, y) = xx' + yy' A) Yes B) No 3) f(x, y) = (x + y)x' A) Yes B) No

functionally complete

Being able to express a Boolean function using only operations from the set given.

minterm

Boolean function whose input variables are v1, v2,...,vk, a product of literals u1u2...uk such that each uj is either vj or vj'.

literal

Boolean variable or the complement of a Boolean variable (for example, x or x').

B

De Morgan's law can be generalized to apply to more than one variable. For example: x+y+w+z=(x'y'w'z')' Use the generalized version of De Morgan's law to find a Boolean expression without addition operations that is equivalent to each DNF expression. x'y'+x'y+xy A) ((x'⋅y')(x'y)(xy))' B) ((x'⋅y')'(x'y)'(xy)')' C) (x'⋅y')'(x'y)'(xy)'

C

De Morgan's law can be generalized to apply to more than one variable. For example: x+y+w+z=(x'y'w'z')' Use the generalized version of De Morgan's law to find a Boolean expression without addition operations that is equivalent to each DNF expression. x'yz'+xy'z+xyz A) (x'yz')'(xy'z)'(xyz)' B) ((x'yz')(xy'z)(xyz))' C) ((x'yz')'(xy'z)'(xyz)')'

Boolean addition

Denoted by +, applies to two elements from {0, 1} and obeys the standard rules for addition, except for 1 + 1.

Boolean multiplication

Denoted by •, applies to two elements from {0, 1} and obeys the standard rules for multiplication.

complement

Denoted with a bar symbol, reverses an element's value.

gates

Electrical devices that build circuits.

Boolean expressions

Expressions that can be built up by applying Boolean operations to Boolean variables or the constants 1 or 0.

OR

Gate that computes Boolean addition.

AND

Gate that computes Boolean multiplication.

Inverter

Gate that computes the complement.

0

Give the value for each Boolean expression. Use the following values for the variables: x = 0, y = 1, z = 1. (x + y')z

0

Give the value for each Boolean expression. Use the following values for the variables: x = 0, y = 1, z = 1. (x + y)(z + 1)'

0

Give the value for each Boolean expression. Use the following values for the variables: x = 0, y = 1, z = 1. (z(x + y))'

1

Give the value for each Boolean expression. Use the following values for the variables: x = 0, y = 1, z = 1. x + yz

satisfiable; unsatisfiable

If there is a way to set the input variables that causes a Boolean expression to evaluate to 1, then the expression is said to be ________. Otherwise, the expression is ________.

Boolean algebra

Is a set of rules and operations for working with variables whose values are either 0 or 1.

Yes.

Is the set {addition, multiplication, complement} functionally complete?

No.

Is the set {addition, multiplication} functionally complete?

Yes.

Is the set {multiplication, complement} functionally complete?

NAND

Operation (which stands for "not and") that is denoted by the symbol ↑. The expression x ↑ y is equivalent to (xy)'.

1) Complement only applied to single variables. 2) No multiplication within a factor.

Rules for CNF expression.

1) Complement only applied to a single variable. 2) No addition within a term.

Rules for DNF Form.

input/output table

Shows the output value of a Boolean function for every possible combination of input values.

digital logic

The area of computer science concerned with designing computer circuitry.

NOR

The operation (which stands for "not or") is denoted by the symbol ↓. The expression x ↓ y is equivalent to x + y.

Combinatorial Circuit

The output of the circuit depends only on the present combination of input values and not on the state of a circuit.

or

The results of the addition operation are the same as the logical ______ operation.

and

The results of the multiplication operation are the same as the logical ______ operation.

1) AND 2) OR 3) Inverter

There are three types of gates. Name them.

equivalent

Two Boolean expressions are ________ if they have the same value for every possible combination of values assigned to the variables contained in the expressions.

x+y=(x'y')'

Two expressions can be added using only multiplication and complement operations by applying De Morgan's law: __________.

xy = (x'+y')'

Two expressions can be multiplied using only addition and complement, written as _______.

Boolean variables

Variables that can have a value of 1 or 0.

a+b+c = (a'b'c')'

What is De Morgan's Law with three terms?

b, a, b, a, a, a.

Which expressions are in conjunctive normal form? 1) z(xy + w') A) Yes B) No 2) (x + y + z)(w' + y') A) Yes B) No 3) (x + y)'(x + y) A) Yes B) No 4) (x + y + z)w' A) Yes B) No 5) x + y + z A) Yes B) No 6) xyw' A) Yes B) No

A, B, A, A, B.

Which expressions are in disjunctive normal form? 1) xy + w' A) Yes B) No 2) (x + y)' A) Yes B) No 3) x + y + z A) Yes B) No 4) xyw' A) Yes B) No 5) (x + y)w' A) Yes B) No

A

Which of the following is an equivalent expression to xy + z that uses only addition and complement? A) (x'+y')'+z B) (x+y)'+z C) ((xy)'z')'

B

Which of the following is an equivalent expression to xy + z that uses only multiplication and complement? A) (x'+y')'+z B) ((xy)'z')' C) (x'y'z')'

It equals to 2.

Why doesn't 1+1 obey the standard rules for addition in Boolean Algebra?

x'

x ↑ x = _______

(xy)'

x ↑ y = ________

(x + y)'

x ↓ y = ________