Boolean Algebra and Logic Gates

• Boolean Algebra is a systematic method for expressing and analyzing the operation of logic circuits.
• George Boole, an English mathematician, introduced Boolean Algebra in the 1850s.
• Claude Shannon applied Boolean algebra to switching circuits in the 1930s, forming the basis of computer logic and design.
• Boolean algebra operates with two values: 0 and 1 (True and False).
• It maps data into bits and bytes, used in telephone switching, computer hardware/software, and electronics.

Applications of Boolean Algebra in Computer

• Designs digital circuits using logic gates such as AND, OR, and NOT.
• Simplifies complex logic circuits.
• This makes circuits smaller, faster, and cheaper.
• Boolean algebra aids in designing circuits for operations like addition and subtraction, using binary numbers (0s and 1s).
• A half-adder circuit, for example, adds two binary numbers.
• Designs control systems that manage how different computer components (memory, CPU, I/O) work together.
• Controls system "on/off" states based on conditions.
• It organizes data storage and access in memory circuits like RAM.
• It designs flip-flops, which are basic memory units that store information (1 or 0).
• It is used in parity checkers and error-correcting codes.
• Parity bits are used to check data transmission accuracy.
• Multiplexers (MUX) select one input from many.
• Demultiplexers (DEMUX) do the reverse, sending one input to multiple outputs.
• If-else statements in programming are Boolean expressions that decide actions based on true/false conditions.
• An example: if (x > 10) checks if x is greater than 10.
• It filters and finds specific data in search algorithms.
• For example, finding records where "age > 30" AND "location = NY".
• Boolean expressions optimize code by minimizing conditions for decision-making and data filtering.

Basic concepts

• Elements that consist of variables or constants and has a value of 0 or 1.
• Operators consisting of AND, OR, and NOT.
• Truth tables are also important

Boolean Variable and Constant

• It's a symbol (A, B, a, b, x, y, z) representing a logical quantity.
• It holds a "true" or "false" value.
• It represents conditions or binary states for decision-making.
• A Boolean constant is a "0" or "1" single-digit binary value which represents a fixed "true" or "false" value in programming.
• Complements are the inverse of variables, indicated by a bar.
• If x=0, then x'=1, and if x=1, then x'=0.

Truth Table

• It systematically lists dependent variable values based on all possible independent variable values.
• It represents input and output conditions for circuits with two or more variables.
• Column count depends on the number of variables in the function.
• An n variable function has 2^n possible combinations.

Logical Operators

• These are used to manipulate Boolean variables and consist of the fundamental logical operators AND, OR and NOT.

AND

• Represented by a dot "." or absence of an operator and performs logical multiplication.
• It lists all combinations of independent variables (A, B) and the result (C) in the equation A.B=C.
• This is used to find commonality between two conditions.
• The symbols are && or simply AND.
• It, in programming, looks like: if(a>b && b>c) then printf("a is largest number");
• The output is "1" only if all independent variables are "1"; otherwise, the output is "0".

OR

• It is represented by "+" and called logical addition.
• It lists all combinations of independent variables (A, B) and the result (C) in the equation A+B=C.
• This is used to find inclusivity or flexibility between two conditions.
• The symbols are || or OR.
• For example: if (temperature < 0 || temperature > 40) then printf("Warning: Temperature is out of the safe range.\n");
• The output is "0" only if all independent variables are "0"; otherwise, the output is "1".

NOT

• Represented by a prime or bar over the variable and performs logical negation or complement.
• This is used to negate a condition.
• Symbols: ! or NOT
• Example: if (!(x > 10)) then printf("x is not greater than 10.\n");

Boolean Laws and Theorems

• Identity Law: A+0=A and A.1=A
• Commutative Law: A+B=B+A and A.B=B.A (order doesn't matter)
• Distributive Law: A.(B+C)=A.B+A.C and A+(B.C)=(A+B).(A+C)
• Complement Law: A + A' = 1 and A . A' = 0
• Idempotent Law: A+A=A and A.A=A
• Null Element: A+1=1 and A.0=0
• Involution/Double Negation Law: (A')' = A
• Associative Law: A+(B+C)= (A+B)+C and A.(B.C)=(A.B).C

De Morgan's Theorem

• Is comprised of two theorems that help simplify complicated logical expressions.

First Theorem

• The complement of a product of variables equals the sum of the complements.
• xy = x + y

Second Theorem

• The complement of a sum of variables equals the product of the complements.
• xy = x + y
• De Morgan's theorems are used in expressions with more than three variables.

Logic Gates

• It's a hardware component and a basic building block for digital circuits that perfoms basic logic functions.
• Seven logic gates are: AND, OR, NOT, NAND, NOR, XOR, XNOR

AND Gate

• Digital circuit that performs an AND operation with two or more inputs and one output.

OR Gate

• Digital circuit that performs an OR operation with two or more inputs and one output.

NOT Gate

• Also called Logical Inverter.
• It reverses the logic state with one input and one output.

NAND Gate

• Combination of NOT with AND operations.
• It operates with has two or more inputs and one output.

NOR Gate

• Combination of NOT with OR operations.
• It has two or more inputs and one output.

XOR Gate

• Exclusive-OR gate.
• The output is 'true' if either, but not all, of the inputs are 'true', other wise the output is 'false'.

XNOR Gate

• Exclusive NOR gate.
• AB=AB+A'B'
• The output is 1 when both inputs are the same (0 or 1) and is the negation of the XOR gate.

Logical Gate Functions Summary

• AND: True if both inputs are true.
• OR: True if at least one input is true.
• NOT: Inverts the input.
• NAND: True unless both inputs are true (negation of AND).
• NOR: True only if both inputs are false (negation of OR).
• XOR: True if inputs are different.
• XNOR: True if inputs are the same (negation of XOR).

Logic Gates Applications

• Combinational Logic Circuits: Used in circuits like adders, multiplexers, and ALUs.
• Control Systems: Manage logic flow in digital systems.
• Microprocessors and Memory: Form arithmetic and logic operations in CPUs and processing units.
• Combining gates creates more complex digital electronics and computation operations.

Boolean Expression

• It is uses mathematical expression that uses variables, constants, and operators to describe a logical relationship.
• The variables can only take the values true (1) or false (0).
• It is the foundation of digital logic, circuits, and design.
• It can be represented in SOP (Sum of Products).

SOP (Sum of Products)

• Boolean expression written as an OR (sum) of multiple ANDed terms (products).
• Each product term is a combination of variables in their normal or complemented form and corresponds to rows in the truth table where the output is 1.
• Combine multiple AND terms (products) using the OR operator (sum).
• AND terms use variables or complements corresponding to rows in the truth table where the function output is 1.
• Expressing it in SOP form means each term is called a minterm which is a product term where all variables appear in true or complemented form, representing a specific input combination where the output is 1, represented as m (0 to 2n-1) where n is the number of variables.
• For 2 variables there will be 22-1=4 minterms mo, m1, m2, m3.

POS (Product of Sums)

• Boolean expression written as an AND (product) of multiple ORed terms (sums).
• Each sum term combines variables, normal or complemented, and corresponds to rows in the truth table where the output is 0.
• Using the AND operator (product) combine multiple OR terms (sums).
• Each OR term is made from their complements corresponding to rows in the truth table where the output is 0.
• The corresponding OR term for the row is included in the product of sums (POS) expression.
• A sum term where all their appears in true or complemented form, representing an input combination where the output is 0, is called a Maxterm as is represented by Mi.

Karnaugh map (K map)

• It's a table used to simplify Boolean algebra expressions and reduces logic gates needed in circuit design.
• The number of cells in k-map is equal to 2, where n is the number of variables involved in the map.

Test your understanding of Boolean algebra laws like commutative, associative, distributive, and DeMorgan's theorems. This quiz also covers logic gates, including AND, OR, NOT, and their applications in digital circuits.