Boolean Algebra Ch.5

combinatorial circuit

In a combinatorial circuit the output of the circuit depends only on the present combination of input values and not on the state of a circuit. Unlike more complex circuitry, combinatorial circuits can not store information over time.

AND gate

The AND gate computes Boolean multiplication

Boolean satisfiability

The Boolean satisfiability 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. 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 satisfiable. Otherwise, the expression is unsatisfiable. A particular assignment of values to the variables satisfies a Boolean expression if the assignment causes the expression to evaluate to 1.

complement

The complement of an element, denoted with a bar symbol, reverse that element's value. Complementing a Boolean value is analogous to applying the ¬ ("not") operation in logic.

Two Boolean expressions are equivalent if...

Two Boolean expressions are equivalent if they have the same value for every possible combination of values assigned to the variables contained in the expressions. In propositional logic, a special symbol (≡) is used to denote logical equivalence. In Boolean algebra, the equal sign (=) is used to denote logical equivalence. The rules of Boolean algebra satisfy the same set of laws (pairs of equivalent expressions) that were satisfied in propositional logic.

Boolean variables

Variables that can have a value of 1 or 0 are called Boolean variables.

OR Gate

computes Boolean addition

Inverter

computes the complement

Boolean addition

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. The results of the addition operation are the same as the logical ∨ ("or") operation.

Boolean multiplication

denoted by •, applies to two elements from {0, 1} and obeys the standard rules for multiplication. The results of the multiplication operation are the same as the logical ∧ ("and") operation.

How would you find a boolean expression that is equivalent to a boolean function defined by an input/output table? (You're given an input/output table. Figure out the boolean function.)

find the rows in which the value of f is 1. Then add them together.

Boolean algebra

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

absorption laws

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

identity laws

x + 0 = x x • 1 = x

domination laws

x + 1 = 1 x • 0 = 0

Idempotent laws

x + x = x x * x = x

complement laws

x + x(bar) = 1 0(bar) = 1 xx(bar) = 0 1(bar) = 0

commutative laws

x + y = y + x xy = yx

distributive laws

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

double complement law

x(bar)(bar) = x

associative laws

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

DeMorgan's laws

(x + y)(bar) = x(bar)y(bar) (xy)(bar) = x(bar) + y(bar)

The steps of circuit design are:

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

precedence rules for Boolean operations

-boolean multiplication takes precedence over boolean addition -the complement operation is applied as soon as the entire expression under that bar is evaluated -parantheses can be used to override the precedence rules

functionally complete

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

boolean expressions

Boolean expressions can be built up by applying Boolean operations to Boolean variables or the constants 1 or 0. The value of a Boolean expression with multiple operations can depend on the order in which the operations are applied. For example the expression 0 • (0 + 1) evaluates to 0, whereas the expression (0 • 0) + 1 evaluates to 1. The rules of precedence for Boolean operations define the correct order to apply the operations.

gates

Circuits are built from electrical devices called gates. 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.