# binary algebra digital logic gates - nebomusic

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Binary_Algebra_Digital_Logic_GatesElectronics and Digital Design Marist School

Binary Algebra

• Binary Systems store “State” or charge • On: 1 • Off: 0

• Binary Equations (Algebra) compare the “state” of a binary equations and return a ‘0’ or ‘1’. • A Binary equation does not compute a numeric

value (like a standard equation). • A Binary equation only returns 0 or 1. • Somewhat like the conditional in an ‘if statement’

or a Boolean function in java.

Examples of Binary/Boolean expressions: F = AB

F = xyz’ + y

F: the result of the equation. The output.

A, B, C, W, x, y, z . . . Variables in the equations representing ‘states’ or truths. The inputs.

Selected Binary Operations Operation Symbol Example Language AND * Or no

symbol F = AB A and B must

be true OR + F = A + B A or B must

be true NOT ‘ sign F = A’ A not true

(not A) NAND ‘ around

and B is not true.

NOR F = (A + B)’ A must not be true or B must not be true.

EOR F = A (+) B A or B must be true but not both A and B true

Truth Tables

• Outline all possibilities of equation inputs and outputs with a given expression. • Number Elements of a binary truth table will equal

to 2 raised to the number of inputs. (2n) • “State Diagram” of the binary system. (All possible

inputs and outputs).

Procedure for creating a truth table. • Make a grid or table with the inputs and outputs. • Columns will be inputs and outputs • Rows will be 2 raised to the number of inputs (2n)

• Fill in the binary count for inputs (0 to 2n-1)

• Use the Binary Equation to calculate the result (output) for each combination of inputs.

Example Creation of Truth Table:

• Equation:

F = AB A B F 0 0 0 0 1 0 1 0 0 1 1 1

A * B = F 0 * 0 = 0 0 * 1 = 0 1 * 0 = 0 1 * 1 = 1

Note the binary counting for A and B.

Another Truth Table example

• Equation: F = x + y

x y F 0 0 0 0 1 1 1 0 1 1 1 1

x + y = F 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1

Truth Table Example

F = xyz + w’ + x’

w x y z F 0 0 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1

Note the number of rows: 2n

Or: 2 raised to 4th power

24 = 16

Row count: 0 to 15 or 0 to (2n – 1)

Algebraic Properties work in similar manners: F = AB + AC -> F = A(B+C)

F = (A+B)(C + D) -> F = AC + AD + BC + BD

F = x(z + y) -> F = xz + xy

Logic Gates and Binary Algebra

• The expressions (AND, OR, NOT, NOR, . . .) in Binary Algebra can be thought of as ‘electronic components’ in a digital system. • We call these ‘components’ : Logic Gates • The Logic Gates take input (on or off signals /

current or no current) and then output a signal based on the configuration of the component. • Combining the Logic Gates and inputs creates a

digital device. (Billions of ‘components’ combined creates what we call a computer).

Table of Logic Gate symbols with Boolean Algebra:

Examples with Logic.ly: F = AB

Example with Logic.ly: F = A + B

Gates can be used to construct

•Decision Circuits (If statements) •Operators (Adders, subtractors . . .) •Memory (Using Latches and Flip Flops) •Counters (Latches or Flip Flops with a clock pulse)

Binary Algebra

• Binary Systems store “State” or charge • On: 1 • Off: 0

• Binary Equations (Algebra) compare the “state” of a binary equations and return a ‘0’ or ‘1’. • A Binary equation does not compute a numeric

value (like a standard equation). • A Binary equation only returns 0 or 1. • Somewhat like the conditional in an ‘if statement’

or a Boolean function in java.

Examples of Binary/Boolean expressions: F = AB

F = xyz’ + y

F: the result of the equation. The output.

A, B, C, W, x, y, z . . . Variables in the equations representing ‘states’ or truths. The inputs.

Selected Binary Operations Operation Symbol Example Language AND * Or no

symbol F = AB A and B must

be true OR + F = A + B A or B must

be true NOT ‘ sign F = A’ A not true

(not A) NAND ‘ around

and B is not true.

NOR F = (A + B)’ A must not be true or B must not be true.

EOR F = A (+) B A or B must be true but not both A and B true

Truth Tables

• Outline all possibilities of equation inputs and outputs with a given expression. • Number Elements of a binary truth table will equal

to 2 raised to the number of inputs. (2n) • “State Diagram” of the binary system. (All possible

inputs and outputs).

Procedure for creating a truth table. • Make a grid or table with the inputs and outputs. • Columns will be inputs and outputs • Rows will be 2 raised to the number of inputs (2n)

• Fill in the binary count for inputs (0 to 2n-1)

• Use the Binary Equation to calculate the result (output) for each combination of inputs.

Example Creation of Truth Table:

• Equation:

F = AB A B F 0 0 0 0 1 0 1 0 0 1 1 1

A * B = F 0 * 0 = 0 0 * 1 = 0 1 * 0 = 0 1 * 1 = 1

Note the binary counting for A and B.

Another Truth Table example

• Equation: F = x + y

x y F 0 0 0 0 1 1 1 0 1 1 1 1

x + y = F 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1

Truth Table Example

F = xyz + w’ + x’

w x y z F 0 0 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1

Note the number of rows: 2n

Or: 2 raised to 4th power

24 = 16

Row count: 0 to 15 or 0 to (2n – 1)

Algebraic Properties work in similar manners: F = AB + AC -> F = A(B+C)

F = (A+B)(C + D) -> F = AC + AD + BC + BD

F = x(z + y) -> F = xz + xy

Logic Gates and Binary Algebra

• The expressions (AND, OR, NOT, NOR, . . .) in Binary Algebra can be thought of as ‘electronic components’ in a digital system. • We call these ‘components’ : Logic Gates • The Logic Gates take input (on or off signals /

current or no current) and then output a signal based on the configuration of the component. • Combining the Logic Gates and inputs creates a

digital device. (Billions of ‘components’ combined creates what we call a computer).

Table of Logic Gate symbols with Boolean Algebra:

Examples with Logic.ly: F = AB

Example with Logic.ly: F = A + B

Gates can be used to construct

•Decision Circuits (If statements) •Operators (Adders, subtractors . . .) •Memory (Using Latches and Flip Flops) •Counters (Latches or Flip Flops with a clock pulse)