Logic Gates

Boolean Algebra and Logic Gates

3.2. Boolean Laws

• Complement Laws

  • Law-9: A + 0 = A

  • Law-10: A . 1 = A

  • Law-11: A + 1 = 1

  • Law-12: A . 0 = 0

  • Law-13: A = A

• Commutative Laws

  • Law-14: A + B = B + A

  • Law-15: A . B = B . A

  • These laws allow for the changing of variable positions in 'OR' and 'AND' expressions without affecting the final result.

• Associative Laws

  • Law-16: A + (B + C) = (A + B) + C

  • Law-17: (A + B) + (C + D) = A + B + C + D

  • Law-18: (B . C) = (A . B) . C

  • Enables the removal of brackets and regrouping of variables.

• Distributive Laws

  • Law-19: A(B + C) = (A . B) + (A . C)

  • Law-20: A + (B . C) = (A + B)(A + C)

  • Permits factoring or multiplying out of an expression.

• Absorptive Laws

  • Law-21: A + (A . B) = A

  • Law-22: A = A

  • Law-23: (A + B) = A . B

• De-Morgan's Laws

  • Law-24: (A + B) = A . B

  • Law-25: A . B = A + B

  • Useful for proving Boolean identities and simplifying expressions using Logic Gates.

3.3. Logic Gates

• Introduction

  • Logic gates serve as the foundational elements of digital electronics and systems.

  • They perform basic logical functions that are fundamental to digital circuits.

• Types of Logic Gates:

  • All gates have one or more inputs and only one output. Output changes based on specific input combinations.

3.3.1. OR Gate

• Functionality: C = A + B

  • Output C is 'true' (1) when either A or B or both inputs are true.

• Truth Table:

  A	B	C (A + B)
  0	0	0
  0	1	1
  1	0	1
  1	1	1

3.3.2. AND Gate

• Functionality: C = A . B

  • Output C is 'true' (1) only when both A and B are true.

• Truth Table:

  A	B	C (A . B)
  0	0	0
  0	1	0
  1	0	0
  1	1	1

3.3.3. NOT Gate

• Functionality: C = A'

  • This single-input gate outputs the inverse of the input.

• Truth Table:

  A	C (NOT A)
  0	1
  1	0

3.3.4. NOR Gate

• Functionality: C = (A + B)'

  • Outputs true only when all inputs are false.

• Truth Table:

  A	B	C (NOR)
  0	0	1
  0	1	0
  1	0	0
  1	1	0

3.3.5. NAND Gate

• Functionality: C = (A . B)'

  • Outputs true unless both inputs are true.

• Truth Table:

  A	B	C (NAND)
  0	0	1
  0	1	1
  1	0	1
  1	1	0

3.3.6. XOR Gate

• Functionality: C = A ⊕ B

  • Outputs true when the inputs are different.

• Truth Table:

  A	B	C (XOR)
  0	0	0
  0	1	1
  1	0	1
  1	1	0

3.3.7. XNOR Gate

• Functionality: C = A ≡ B

  • Outputs true when the inputs are the same.

• Truth Table:

  A	B	C (XNOR)
  0	0	1
  0	1	0
  1	0	0
  1	1	1

3.3.8. Bubbled Gate

• Definition: A bubbled gate indicates that inputs are negated (inverted) before processing.

3.4. Interconnecting Gates

• Gates can be interconnected to form complex circuits or systems, referred to as 'Gating Networks'.

3.5. Boolean Functions

• Definition: A digital signal representing two states (High/0 or Low/1).

• Example: X = A . B + C(D + E) is a Boolean expression with variables A, B, C, D, and E.

3.6. Duality Theorem

• The dual of a Boolean expression can be found by swapping ANDs with ORs and inverting 0s and 1s.

• Example: From A + 0 = A, the dual is A . 1 = A.

3.7. De-Morgan's Theorems

• Theorems:

  1. (A + B)' = A' . B'

  2. (A . B)' = A' + B'

• Useful for simplifying complex expressions.