Second Semester_Digital Logic_Logic Gates and Boolean Algebra

 

2.1 Basic Definition of Boolean Algebra

• Boolean Algebra: A mathematical system for analyzing and simplifying logic circuits.

• Values are binary: 0 (False) and 1 (True).

• Operations follow specific rules and laws.

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2.2 Basic Theory of Boolean Algebra, Boolean Functions, Logical Operations

• Boolean Functions: Functions that take binary inputs and produce a binary output.

  • Example: F(A, B) = A·B + A'·B

• Logical Operations:

  1. AND (·): Output 1 if all inputs are 1.

  2. OR (+): Output 1 if at least one input is 1.

  3. NOT (’): Output is the complement of input.

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2.3 Logic Gates, IC Digital Logic Families. Basic Gates

• Logic Gates: Hardware implementation of Boolean operations.

  • AND, OR, NOT gates

• Digital Logic Families: Types of IC technologies:

  • TTL (Transistor-Transistor Logic)

  • CMOS (Complementary MOS)

  • ECL (Emitter-Coupled Logic)

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2.4 Universal Gates (NAND and NOR), Other Gates (XOR, XNOR)

• Universal Gates: Can implement any Boolean function.

  • NAND gate

  • NOR gate

• Other Gates:

  • XOR (Exclusive OR): Output 1 if inputs are different.

  • XNOR (Exclusive NOR): Output 1 if inputs are the same.

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2.5 Boolean Identities, De Morgan’s Laws

• Boolean Identities: Rules to simplify expressions. Examples:

  • A + 0 = A, A·1 = A, A + A = A, A·A = A

• De Morgan’s Laws: Convert AND ↔ OR with complements:

1. (A·B)’ = A’ + B’

2. (A + B)’ = A’·B’

🔹 Basic Logic Gates

1. NOT Gate (Inverter)

• Equation: $Y = \overline{A}$

• Truth Table:

  • $A=0 \Rightarrow Y=1$

  • $A=1 \Rightarrow Y=0$

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2. AND Gate

• Equation: $Y = A \cdot B$ (sometimes written as $AB$)

• Truth Table:

  • $0 \cdot 0 = 0$

  • $0 \cdot 1 = 0$

  • $1 \cdot 0 = 0$

  • $1 \cdot 1 = 1$

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3. OR Gate

• Equation: $Y = A + B$

• Truth Table:

  • $0+0 = 0$

  • $0+1 = 1$

  • $1+0 = 1$

  • $1+1 = 1$

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🔹 Universal Gates

4. NAND Gate (NOT-AND)

• Equation: $Y = \overline{A \cdot B}$

• Truth Table:

  • $0 \cdot 0 = 0 \Rightarrow 1$

  • $0 \cdot 1 = 0 \Rightarrow 1$

  • $1 \cdot 0 = 0 \Rightarrow 1$

  • $1 \cdot 1 = 1 \Rightarrow 0$

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5. NOR Gate (NOT-OR)

• Equation: $Y = \overline{A + B}$

• Truth Table:

  • $0+0 = 0 \Rightarrow 1$

  • $0+1 = 1 \Rightarrow 0$

  • $1+0 = 1 \Rightarrow 0$

  • $1+1 = 1 \Rightarrow 0$

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🔹 Exclusive Gates

6. XOR (Exclusive-OR)

• Equation: $Y = A \oplus B = A\overline{B} + \overline{A}B$

• Truth Table:

  • $0 \oplus 0 = 0$

  • $0 \oplus 1 = 1$

  • $1 \oplus 0 = 1$

  • $1 \oplus 1 = 0$

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7. XNOR (Exclusive-NOR / Equivalence)

• Equation: $Y = \overline{A \oplus B} = AB + \overline{A}\,\overline{B}$

• Truth Table:

  • $0 \text{ XNOR } 0 = 1$

  • $0 \text{ XNOR } 1 = 0$

  • $1 \text{ XNOR } 0 = 0$

  • $1 \text{ XNOR } 1 = 1$

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✅ Summary Table (Equations Only)

Gate	Symbol	Equation
NOT	$\overline{A}$	Inverts input
AND	$A \cdot B$	Outputs 1 if both are 1
OR	$A + B$	Outputs 1 if any is 1
NAND	$\overline{A \cdot B}$	Inverse of AND
NOR	$\overline{A + B}$	Inverse of OR
XOR	$A \oplus B = A\overline{B} + \overline{A}B$	Outputs 1 if inputs differ
XNOR	$\overline{A \oplus B} = AB + \overline{A}\,\overline{B}$	Outputs 1 if inputs are same