Chapter 2: Number System and Logic Functions

Chapter 2: Number System and Logic Functions

2.1 Number System and Conversion

2.1.1 Decimal, Binary, Octal, Hexadecimal Number System & Conversion

Number Systems:

1. Decimal (Base 10):

   • Uses digits 0-9.
   • Each position represents a power of 10 (e.g., $345 = 3 \times 10^2 + 4 \times 10^1 + 5 \times 10^0$).
   • Conversion: To convert from decimal to binary, octal, or hexadecimal, divide the number by the base and record the remainders.

2. Binary (Base 2):

   • Uses digits 0 and 1.
   • Each position represents a power of 2 (e.g., $1011 = 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 11_{10}$).
   • Conversion: Decimal to binary involves dividing by 2 and noting remainders.

3. Octal (Base 8):

   • Uses digits 0-7.
   • Each position represents a power of 8 (e.g., $27 = 2 \times 8^1 + 7 \times 8^0 = 23_{10}$).
   • Conversion: Decimal to octal is similar to binary but with base 8.

4. Hexadecimal (Base 16):

   • Uses digits 0-9 and letters A-F (A=10, B=11, C=12, D=13, E=14, F=15).
   • Each position represents a power of 16 (e.g., $1A3 = 1 \times 16^2 + 10 \times 16^1 + 3 \times 16^0 = 419_{10}$).
   • Conversion: Decimal to hexadecimal involves dividing by 16.

2.1.2 Calculation in Binary: Addition and Subtraction

• Binary Addition:

  • Rules:

    • $0 + 0 = 0$
    • $0 + 1 = 1$
    • $1 + 0 = 1$
    • $1 + 1 = 0$ (carry 1 to the next higher bit)
    • $1 + 1 + 1 = 1$ (carry 1 to the next higher bit)

  • Example: $1011 + 1101$

    sql
    Copy code
        1011
      + 1101
      ------
       11000 (binary) = 24 (decimal)
    

• Binary Subtraction:

  • Rules:
    • $0 - 0 = 0$
    • $1 - 0 = 1$
    • $1 - 1 = 0$
    • $0 - 1$: requires borrowing.
  • Example: $1100 - 1010$

    sql
    Copy code
         1100
      -  1010
      -------
         0100 (binary) = 4 (decimal)
    

2.1.3 One’s and Two’s Complement Methods of Binary Subtraction

• One’s Complement:

  • Involves flipping all bits (0s become 1s and vice versa).
  • Example: One’s complement of $1010$ is $0101$.

• Two’s Complement:

  • One’s complement plus one.
  • Used for binary subtraction.
  • Example: Two’s complement of $1010$:
    • One’s complement: $0101$
    • Adding one: $0101 + 0001 = 0110$ (Two’s complement = $0110$).

• Subtraction Example:

  • To subtract $A - B$, convert $B$ to its two’s complement and add it to $A$.
  • If there's a carry, discard it.

2.2 Logic Function and Boolean Algebra

2.2.1 Introduction to Boolean Algebra

• Boolean Algebra:
  • A branch of algebra that involves binary variables and logical operations.
  • Fundamental to digital circuits and computer logic.
  • Uses two values: True (1) and False (0).

2.2.2 Introduction to Boolean Values, Truth Table, Boolean Expression, and Boolean Function

• Boolean Values: Represent logical statements as either true (1) or false (0).

• Truth Table: A table that lists all possible values of variables and the corresponding output of a Boolean expression.

  • Example: Truth table for $A$ AND $B$:

    A	B	A AND B
    0	0	0
    0	1	0
    1	0	0
    1	1	1

• Boolean Expression: Represents a logical relationship using variables and operators (AND, OR, NOT).

  • Example: $A + B$ represents $A$ OR $B$.

• Boolean Function: A function that takes Boolean inputs and produces a Boolean output.

2.2.3 Logic Gates

• AND Gate:

  • Definition: Outputs true only if all inputs are true.
  • Truth Table:

    A	B	A AND B
    0	0	0
    0	1	0
    1	0	0
    1	1	1

  • Logic Symbol:

• OR Gate:

  • Definition: Outputs true if at least one input is true.
  • Truth Table:

    A	B	A OR B
    0	0	0
    0	1	1
    1	0	1
    1	1	1

  • Logic Symbol:

• NOT Gate:

  • Definition: Outputs the opposite of the input.
  • Truth Table:

    A	NOT A
    0	1
    1	0

  • Logic Symbol:

• NAND Gate:

  • Definition: Outputs false only if all inputs are true.
  • Truth Table:

    A	B	A NAND B
    0	0	1
    0	1	1
    1	0	1
    1	1	0

  • Logic Symbol:

• NOR Gate:

  • Definition: Outputs true only if all inputs are false.
  • Truth Table:

    A	B	A NOR B
    0	0	1
    0	1	0
    1	0	0
    1	1	0

  • Logic Symbol:

• XOR Gate:

  • Definition: Outputs true if the inputs are different.
  • Truth Table:

    A	B	A XOR B
    0	0	0
    0	1	1
    1	0	1
    1	1	0

  • Logic Symbol:

• XNOR Gate:

  • Definition: Outputs true if the inputs are the same.
  • Truth Table:

    A	B	A XNOR B
    0	0	1
    0	1	0
    1	0	0
    1	1	1

  • Logic Symbol:

2.2.4 Laws of Boolean Algebra

• Boolean Identities:
  • Complement Laws:
    • $A + \overline{A} = 1$
    • $A \cdot \overline{A} = 0$
  • Identity Laws:
    • $A + 0 = A$
    • $A \cdot 1 = A$
  • Commutative Laws:
    • $A + B = B + A$
    • $A \cdot B = B \cdot A$
  • Associative Laws:
    • $A + (B + C) = (A + B) + C$
    • $A \cdot (B \cdot C) = (A \cdot B) \cdot C$
  • Distributive Law:
    • $A \cdot (B + C) = A \cdot B + A \cdot C$

2.2.5 Statement and Verification of Laws of Boolean Algebra Using Truth Table

• Verification:

  • Truth tables can be constructed for both sides of the equation for each law to confirm their validity.

  Example: Verification of the Commutative Law:

  • For $A + B$ and $B + A$:

    A	B	A + B	B + A
    0	0	0	0
    0	1	1	1
    1	0	1	1
    1	1	1	1

• Both outputs match, proving $A + B = B + A$.