Logic Gates & Boolean Algebra Chin-Sung Lin Eleanor Roosevelt High School

Logic Gates & Boolean Algebra System Concept Systems & Subsystems Analog Systems vs. Digital Systems Combinational vs. Sequential Circuits Truth Tables & Basic Logic Gates Logic Simulation Digital Logic Circuits Equivalent Logic Circuits Digital Building Blocks

System Concept

What Are They In Common? VS. Computer Onion

System Concept System (System Function) Input Output

System Concept System (System Function) Input Output Output

System Concept System (System Function) Input Output A convenient way to view and understand both the nature and man-made worlds. Hide details and complexity from the viewers by encapsulating this detail information into a “system”. Only the inputs/outputs (I/O) of the system, and the function of the system (called system function) are important.

Systems & Subsystems

Systems & Subsystems System Input Output System (System Function) Input Output Systems can be further divided down to subsystems. Subsystems are connected together through inputs and outputs to form the larger system.

Systems & Subsystems Subsystem Sub-system Input Output Input Output Input Output Subsystem themselves can also be further divided down to “Sub-subsystems”. This process can be continued until we reach the most basic elements of the digital logic world— basic logic gates. A system has a layered hierarchical structure like an onion.

Systems & Subsystems

System & Subsystem Example

System & Subsystem Example

Analog System vs. Digital Systems

Analog System vs. Digital Systems Input Output Digital System Input Output Two types of systems: Analog system and digital system.

Analog System vs. Digital Systems Input Output The values of input/output/internal signals of an analog system can vary over a continuous range of values.

Analog System vs. Digital Systems Input Output 1 1 The values of input/output/internal signals of a digital system can only be 1’s and 0’s.

Analog System vs. Digital Systems Digital I/P 1 A/D Converter Analog I/P Lots of real-world signals are analog in nature. Analog-to-Digital (A/D) converter has been used to process (digitization/quantization) the incoming analog signals and change them to digital (binary) signals. The digital signals can now be processed by the digital system (e.g., microprocessor).

Analog System vs. Digital Systems Digital I/P Digital O/P 1 D/A Converter A/D Analog I/P Analog O/P The output digital signals from the digital system are digital (binary) in nature. Digital-to-Analog (D/A) converter has been used to process the outgoing digital signals and change them to analog (continuous) signals. Most of the computers are the mix of analog and digital systems.

Analog System vs. Digital Systems Digital I/P Digital O/P 1 D/A Converter A/D Analog I/P Analog O/P

Digital Systems Digital System Digital I/P Digital O/P 1 D/A Converter A/D Analog I/P Analog O/P In this unit, we are going to focus ONLY on the digital system.

Combinational vs. Sequential Systems

Combinational vs. Sequential Systems Input Output 1 Sequential System Input Output 1 Two types of digital systems/circuits— combinational and sequential.

Combinational vs. Sequential Systems Input Output 1 Combinational System Input Output 1 Combinational System Input Output 1 The outputs of a combinational digital system/circuits are solely decided by its inputs.

Combinational vs. Sequential Systems Input Output 1 Sequential System Input Output 1 Sequential System Input Output 1 The outputs of a sequential digital system/circuits depend not only on its inputs, but also on the “current state” of the system.

Combinational vs. Sequential Systems Input Sequential System State: 101 State: 110 State: 001 Output 1 1 1 1 1 Input Output 1 1 1 1 1 Input Output 1 1 1 1 The outputs of a sequential digital system/circuits depend not only on its inputs, but also on the “current state” of the system. Sequential system/circuits have some sort of “memory” in it, so it can “memorize” the “current state” of the system and behave accordingly.

Combinational vs. Sequential Systems Input Output 1 Sequential System Input Output 1

Combinational Systems Input Output 1 Sequential System Input Output 1 In this unit, we are going to focus MAINLY on the combinational digital system.

Truth Tables & Basic Logic Gates

Truth Tables p q pq 1 A truth table shows how a logic circuit's output responds to various combinations of the inputs. A truth table describe the system function of a logic system. Use “1” and “0” to represent “T” and “F” respectively. Use a logic gate symbol to represent the function.

Truth Tables & Basic Logic Gates X = AB There are eight basic logic gates. However, only AND, OR and NOT are the most fundamental ones.

Chapter 1 - Essential of Geometry Negations (NOT) The negation of a statement always has the opposite truth value of the original statement and is usually formed by adding the word not to the given statement. Statement Right angle is 90o TRUE Negation Right angle is not 90o FALSE Statement Triangle has 4 sides FALSE Negation Triangle does not have 4 sides TRUE

Truth Table – Negation (NOT) Chapter 1 - Essential of Geometry Truth Table – Negation (NOT) The relationship between a statement p and its negation ~p can be summarized in a truth table. A statement p and its negation ~p have opposite truth values. p ~p 1

Chapter 1 - Essential of Geometry Conjunctions (AND) A compound statement formed by combining two simple statements using the word and. Statement: p, q Conjunction: p and q Symbols: p ^ q

Truth Table – Conjunctions (AND) Chapter 1 - Essential of Geometry Truth Table – Conjunctions (AND) A conjunction is true when both statements are true. When one or both statements are false, the conjunction is false. p q p ^ q 1

Chapter 1 - Essential of Geometry Disjunctions (OR) A compound statement formed by combining two simple statements using the word or. Statement: p, q Disjunction p or q Symbols: p V q

Truth Table – Disjunctions (OR) Chapter 1 - Essential of Geometry Truth Table – Disjunctions (OR) A disjunction is true when one or both statements are true. When both statements are false, the disjunction is false. p q p V q 1

Truth Tables & Basic Logic Gates System Function Input X Output Y 0/1 1/0 X Y 1 The truth table summarizes all the possible values of input signals and their corresponding output signal values. The truth table is an effective way to describe the system function of a digital system.

Truth Tables & Basic Logic Gates X 0/1 1/0 A logic gate is a device performing a logical operation on one or more logical inputs, and produces a single logical output.

Boolean Algebra & Boolean Functions Logic Gate X 0/1 1/0 Boolean algebra is a branch of algebra in which the values of the variables are the truth values true (1) and false (0). The main operations of Boolean algebra are the conjunction (and, ), the disjunction (or, +), and the negation (not, ).

Boolean Algebra & Boolean Functions Y = X Y = A  B Y = A + B (NOT) (AND) (OR) A Boolean function describes how to determine a Boolean value output based on some logical calculation from Boolean inputs.

Truth Tables & Basic Logic Gates Boolean Function Truth Table Logic Gate Y = X X Y 1 X NOT (Inverter)

Truth Tables & Basic Logic Gates Boolean Function Truth Table Logic Gate Y = A  B A B Y 1 AND

Truth Tables & Basic Logic Gates Boolean Function Truth Table Logic Gate Y = A + B A B Y 1 OR

Truth Tables & Basic Logic Gates Boolean Function Truth Table Logic Gate Y = X X Y 1 X Buffer

Truth Tables & Basic Logic Gates Boolean Function Truth Table Logic Gate Y = A  B A B Y 1 NAND

Truth Tables & Basic Logic Gates Boolean Function Truth Table Logic Gate Y = A + B A B Y 1 NOR

Truth Tables & Basic Logic Gates Boolean Function Truth Table Logic Gate Y = A ⊕ B A B Y 1 XOR

Truth Tables & Basic Logic Gates Boolean Function Truth Table Logic Gate Y = A  B A B Y 1 Y = A ⊕ B XNOR

Truth Tables & Basic Logic Gates X X

Truth Tables & Basic Logic Gates X X

Truth Tables & Basic Logic Gates X X

Truth Tables & Basic Logic Gates X = AB

Digital Logic Circuits

Digital Logic Circuits Y Logic gates can be connected and cascaded to form logic circuits. Every logic circuits can be treated as a system, and can be described by a system function— truth table. We can derive the truth table of the circuit by evaluating the logic circuit stage by stage.

Digital Logic Circuits E Y F D A B C D E F Y 1

Digital Logic Circuits E Y F D A B C D E F Y 1

Digital Logic Circuits E Y F D A B C D E F Y 1

Digital Logic Circuits E Y F D A B C D E F Y 1

Digital Logic Circuits E Y F D A B C D E F Y 1

Digital Logic Circuits E Y F D A B C D E F Y 1

Digital Logic Circuits E Y = A  B + F D A B C D E F Y 1

Digital Logic Circuits Exercise F G A B C D E F G Q 1

Digital Logic Circuits Exercise F G A B C D E F G Q 1

Logic Simulation

Logic Simulation Logic simulation is the use of simulation software (Logisim) to predict the behavior of digital circuits.

Equivalent Logic Circuits

Equivalent Logic Circuits Logic circuits of different gates and forms can have an identical truth table. These circuits are called equivalent logic circuits. This implies that a digital system with certain system function (truth table) can have many different implementations.

Laws of Boolean Algebra One variable NOT: x = x AND: x · x = x x · x = 0 OR: x + x = x x + x = 1 XOR: x ⊕ x = 0 x ⊕ x = 1

Laws of Boolean Algebra Commutativity AND: x · y = y · x OR: x + y = y + x XOR: x ⊕ y = y ⊕ x Associativity AND: (x · y) · z = x · (y · z) OR: (x + y) + z = x + (y + z) XOR: (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)

Laws of Boolean Algebra Distributivity x · (y + z) = (x · y) + (x · z) x + (y · z) = (x + y) · (x + z) x · (y ⊕ z) = (x · y) ⊕ (x · z) De Morgan’s laws NAND: x · y = x + y NOR: x + y = x · y

Logic Simplification We can apply the laws of Boolean algebra to reduce the expression to its simplest form (simplest defined as requiring the fewest gates to implement)

Equivalent Logic Circuits Different implementations are chosen to meet different design considerations Less number of gates (area) less kind of gates (gate type), and less stages of the circuits (speed).

Equivalent Logic Circuits For example, what are the following logic circuits?

Equivalent Logic Circuits For example, all of the following four logic circuits have the same truth table, and implement the XOR gate functionality.

Digital Building Blocks

Digital Building Blocks All these logic circuits can be encapsulated into a block (XOR gate) and treated as a system.

Digital Building Blocks This XOR gate can be further used as a building block to build larger and more complicated logic circuit such as a “full adder”.

Digital Building Blocks A full adder can again be treated as a building block to build a larger logic circuit called Arithmetic-Logic Unit (ALU).

Digital Building Blocks ALU can again be treated as a building block to build a Central Processing Unit (CPU).

Digital Building Blocks This process can go on and on to build an Intel® Intel® Core™ i7 Processor with these digital logic building blocks. Intel Core i7 Processor

Digital Building Blocks The processor is so complicated at this level that it can contain hundreds of millions of basic logic gates.

Logic Gates & Boolean Algebra X X