14.1 Boolean algebra and logic operations

Boolean algebra is the math behind digital logic, using binary variables and operations like AND, OR, and NOT. It's the foundation for understanding how computers process information and make decisions.

Logic operations form the building blocks of digital circuits. From basic AND and OR gates to more complex XOR and NAND operations, these tools let us create the intricate systems that power modern technology.

Boolean Algebra Fundamentals

Boolean Algebra and Variables

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• Boolean algebra is a branch of mathematics that deals with the manipulation of logical expressions involving binary variables
• Uses logical operators such as AND, OR, and NOT to perform operations on binary variables
• Boolean variables can only take on two possible values: 0 (false) or 1 (true)
• These variables are often used to represent the state of a digital system or the truth value of a logical proposition

Boolean Functions and Expressions

• Boolean functions are mathematical expressions that take Boolean variables as inputs and produce a Boolean output
• Can be represented using truth tables, which list all possible combinations of input values and their corresponding output values
• Boolean expressions are combinations of Boolean variables and operators that evaluate to either 0 or 1
• Examples of Boolean expressions include $A \cdot B$ (A AND B) and $A + B$ (A OR B)

Laws of Boolean Algebra

• Commutative law states that the order of operands does not affect the result of an operation
  • For AND: $A \cdot B = B \cdot A$
  • For OR: $A + B = B + A$
• Associative law allows for the grouping of operands in any order without affecting the result
  • For AND: $(A \cdot B) \cdot C = A \cdot (B \cdot C)$
  • For OR: $(A + B) + C = A + (B + C)$
• Distributive law allows for the distribution of an operator over another
  • AND distributes over OR: $A \cdot (B + C) = (A \cdot B) + (A \cdot C)$
  • OR distributes over AND: $A + (B \cdot C) = (A + B) \cdot (A + C)$

Basic Logic Operations

AND Operation

• AND operation returns 1 (true) only if all inputs are 1 (true)
• Denoted by the symbol $\cdot$ or the word AND
• Truth table for AND:
  • 0 AND 0 = 0
  • 0 AND 1 = 0
  • 1 AND 0 = 0
  • 1 AND 1 = 1
• Can be implemented using an AND gate in digital circuits

OR Operation

• OR operation returns 1 (true) if at least one input is 1 (true)
• Denoted by the symbol $+$ or the word OR
• Truth table for OR:
  • 0 OR 0 = 0
  • 0 OR 1 = 1
  • 1 OR 0 = 1
  • 1 OR 1 = 1
• Can be implemented using an OR gate in digital circuits

NOT Operation

• NOT operation, also known as inversion, returns the opposite of the input value
• Denoted by the symbol $\bar{}$ or the word NOT
• Truth table for NOT:
  • NOT 0 = 1
  • NOT 1 = 0
• Can be implemented using a NOT gate or inverter in digital circuits

Advanced Logic Operations

XOR Operation

• XOR (exclusive OR) operation returns 1 (true) if exactly one input is 1 (true)
• Denoted by the symbol $\oplus$ or the word XOR
• Truth table for XOR:
  • 0 XOR 0 = 0
  • 0 XOR 1 = 1
  • 1 XOR 0 = 1
  • 1 XOR 1 = 0
• Can be implemented using an XOR gate in digital circuits
• Used in error detection and correction schemes, such as parity checks

NAND and NOR Operations

• NAND (NOT AND) operation returns the opposite of the AND operation
  • Denoted by the symbol $\overline{\cdot}$ or the word NAND
  • Truth table for NAND:
    • 0 NAND 0 = 1
    • 0 NAND 1 = 1
    • 1 NAND 0 = 1
    • 1 NAND 1 = 0
  • Can be implemented using a NAND gate in digital circuits
• NOR (NOT OR) operation returns the opposite of the OR operation
  • Denoted by the symbol $\overline{+}$ or the word NOR
  • Truth table for NOR:
    • 0 NOR 0 = 1
    • 0 NOR 1 = 0
    • 1 NOR 0 = 0
    • 1 NOR 1 = 0
  • Can be implemented using a NOR gate in digital circuits
• NAND and NOR gates are considered universal gates because any Boolean function can be implemented using only NAND or only NOR gates