CSCI 241 - Ch. 4 Boolean Algebra

Precedence rules for Boolean operations

1. Boolean multiplication takes precedence over Boolean addition. 2. The complement operation is applied as soon as the entire expression under the bar is evaluated. 3. Parentheses can be used to override the precedence rules.

Steps in the circuit design process

1. Build an input/output table with the desired output for every possible combination of values for the input variables. 2. Construct a Boolean expression that computes the same function as the function specified in the input/output table. 3. Construct a digital circuit that realizes the Boolean expression.

A Boolean function with k input variables will require an input/output table with ______ rows.

2^k

Showing that an expression is unsatisfiable by considering all possible settings for the variables takes time proportional to ________ for an expression with n variables.

2^n

The ________ gate computes Boolean multiplication

AND

x + (xy) = x or x(x + y) = x

Absorption laws

(x + y) + z = x + (y + z) or (xy)z = x(yz)

Associative laws

Denoted by +, applies to two elements from {0, 1} and obeys the standard rules for addition, except for 1 + 1 (an outcome of 2 would not be allowed because all values in Boolean algebra must be 0 or 1).

Boolean addition

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

Boolean algebra

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

Boolean expression

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

Boolean multiplication

The ________ __________ problem (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.

Boolean satisfiability

Variables that can have a value of 1 or 0

Boolean variables

x + y = y + x or xy = yx

Commutative laws

x + yz =(x + y)(x + z) or x(y + z) =xy + xz

Distributive laws

x + 1 = 1 or x • 0 = 0

Domination laws

x + x = x or x • x = x

Idempotent laws

x + 0 = x or x • 1 = x

Identity laws

A product term in a Boolean expression where all possible variables appear once in true or complement form.

Minterm

The Boolean expression for the function specified in an input/output table is a sum of minterms. The minterms for which rows are included in sum?

Minterms are included for the rows in which the function evaluates to 1

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

NAND

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

NOR

The ______ gate computes Boolean addition

OR

Input/output tables for NAND and NOR

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Boolean addition and logical ∨

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Boolean complement and logical ¬

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Boolean multiplication and logical ∧

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Complement laws

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De Morgan's laws

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Double complement law

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To find an equivalent Boolean expression for function f expressed by a table, first find the rows in which the value of f is 1. Now add all the terms together.

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Here is a method to express an arbitrary Boolean function using only multiplication and complement operations:

Start with the input/output table for the function. Find a DNF expression that is equivalent to the function. Repeatedly apply De Morgan's law to eliminate each addition operation.

A CNF expression can only apply the addition operation to single literals.

T

A gate receives some number of Boolean input values and produces an output based on the values of the inputs. Thus, a gate implements a simple Boolean function.

T

A minterm has to include a literal for each variable in a function.

T

A term in a DNF expression can be a single literal, such as w' (the complement of w)

T

Boolean Functions: Let B = {0, 1}. Then B^k is the set of all k-tuples over the set B. A Boolean function with k input variables maps B^k to B.

T

Every combinatorial circuit computes a Boolean function

T

If a Boolean expression is in DNF form, it is very easy to determine if the expression is satisfiable. The formula is satisfiable if and only if there is a term that does not contain a variable and its negation.

T

In a CNF expression, addition can only be applied to literals. In particular, an expression in conjunctive normal form can not contain an expression of the form xy + z because xy is not a literal.

T

In a DNF expression, multiplication can only be applied to literals. correct: xyw' not correct: (x + y)w'

T

In conjunctive normal form, the complement operation can only be applied to single variables.

T

In disjunctive normal form, the complement operation can only be applied to single variables.

T

In propositional logic, a special symbol (≡) is used to denote logical equivalence. In Boolean algebra, the equal sign (=) is used to denote logical equivalence.

T

It is much more difficult to determine if a CNF expression is satisfiable than it is for a DNF expression. Each clause in the CNF represents a constraint that must be satisfied. Typically each constraint can be expressed as a Boolean sum of literals. An individual clause is satisfied by an assignment of values to the variables if the assignment makes the clause evaluate to 1. The entire expression is a product of all the clauses. Therefore the expression evaluates to 1 if and only if all the clauses are satisfied.

T

More complex functions can be computed by connecting gates together. The output of one gate can feed into the input of another gate. The output of a gate can also branch and feed into multiple gates.

T

Neither addition, multiplication, nor complement are by themselves functionally complete, so only a new operation could possibly be functionally complete by itself.

T

The clauses in a CNF expression can be single literals, such as w

T

The definitions for Boolean operations are basically a restatement of logical operations with slightly different notation.

T

The expression x + y + z is the same as (x + y + z). A CNF expression can have only one clause as long as that single clause is a sum of literals.

T

The set {addition, multiplication, complement} is functionally complete because any Boolean function can be expressed in disjunctive normal form which only uses addition, multiplication, and complement operations.

T

The set {addition, multiplication} is not functionally complete because there is no way to compute x' using only addition and multiplications.

T

The shape of a gate indicates the particular operation that gate computes.

T

Two expressions can be multiplied using only addition and complement by applying De Morgan's law: xy = (x'+y')'

T

Using a table to define a Boolean function is only feasible for functions with a small number of input variables. Alternatively, one could use a Boolean expression to define a function.

T

The value 1 corresponds to ____ in the rules of propositional logic, and the value 0 corresponds to ____.

T , F

What are the three types of gates that perform the three basic Boolean operations?

The AND gate, the OR gate, and the inverter

Is the set {multiplication, complement} functionally complete?

Two expressions can be added using only multiplication and complement operations by applying De Morgan's law: x+y = (x'⋅y')'

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

a literal

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

and

Each term in the product that is a sum of literals is called a ________

clause

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

combinatorial

The __________ of an element, denoted with a bar symbol, reverses that element's value.

complement

A Boolean expression that is a product of sums of literals

conjunctive normal form

The area of computer science concerned with designing computer circuitry is called _________ _________

digital logic

A Boolean expression that is a sum of products of literals

disjunctive normal form

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

equivalent

A Boolean _________ maps one or more Boolean input values to the set {0, 1}

function

A set of operations is __________ _________ if any Boolean function can be expressed using only operations from the set.

functionally complete

Circuits are built from electrical devices called _________.

gates

One way to define a Boolean function is to provide an ___________ that shows the output value of the function for every possible combination of input values

input/output table

The __________ gate computes the complement

inverter

Complementing a Boolean value is analogous to applying the _______ operation in logic.

not

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

or

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 __________.

satisfiable, unsatisfiable

Constructing a Boolean expression from a circuit

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Constructing a circuit from a Boolean expression

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Is this CNF? xyw'

yes