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Realization of Boolean Expression using Gates

Introduction

When we realize a Boolean expression, we take the mathematical representation of logic (Boolean expression) and convert it into a digital circuit using basic logic gates (AND, OR, and NOT). This is important for building physical systems like computers, where circuits are constructed based on Boolean logic.

🛠 Logic Gates Overview

  1. AND Gate:
    • Output is 1 only when both inputs are 1.
    • Symbol: Aâ‹…BA \cdot BAâ‹…B
    • Truth Table: ABA · B000010100111
  2. OR Gate:
    • Output is 1 when any one or both inputs are 1.
    • Symbol: A+BA + BA+B
    • Truth Table: ABA + B000011101111
  3. NOT Gate:
    • Output is the inverse of the input.
    • Symbol: A‾\overline{A}A
    • Truth Table: AA‾\overline{A}A0110

🔥 Step-by-Step Realization Process

Let’s now break down the steps for realizing Boolean expressions using gates.

Example 1: Realization of SOP (Sum of Products) Expression

Boolean Expression: F=A⋅B+C‾F = A \cdot B + \overline{C}F=A⋅B+C

Step-by-Step Realization:

  1. Identify the parts of the expression:
    • First part: Aâ‹…BA \cdot BAâ‹…B — This is an AND operation between A and B.
    • Second part: C‾\overline{C}C — This is a NOT operation applied to C.
  2. Realize the first part:
    • Use an AND gate for AAA and BBB.
    • The output of the AND gate will be Aâ‹…BA \cdot BAâ‹…B.
  3. Realize the second part:
    • Use a NOT gate for CCC.
    • The output of the NOT gate will be C‾\overline{C}C.
  4. Final Step:
    • Use an OR gate to combine Aâ‹…BA \cdot BAâ‹…B and C‾\overline{C}C.
    • The final output is F=(Aâ‹…B)+C‾F = (A \cdot B) + \overline{C}F=(Aâ‹…B)+C.

Logic Gate Diagram:

  • Inputs: A, B, and C
  • Gates:
    • One AND gate for Aâ‹…BA \cdot BAâ‹…B
    • One NOT gate for C‾\overline{C}C
    • One OR gate for (Aâ‹…B)+C‾(A \cdot B) + \overline{C}(Aâ‹…B)+C

Circuit Diagram Overview:

  1. AND Gate: Inputs are A and B; output is Aâ‹…BA \cdot BAâ‹…B.
  2. NOT Gate: Input is C; output is C‾\overline{C}C.
  3. OR Gate: Inputs are A⋅BA \cdot BA⋅B and C‾\overline{C}C; output is the final function FFF.

Example 2: Realization of POS (Product of Sums) Expression

Boolean Expression: F=(A+B)⋅(C‾+D)F = (A + B) \cdot (\overline{C} + D)F=(A+B)⋅(C+D)

Step-by-Step Realization:

  1. Identify the parts of the expression:
    • First part: A+BA + BA+B — This is an OR operation between A and B.
    • Second part: C‾+D\overline{C} + DC+D — This is an OR operation between NOT C and D.
  2. Realize the first part:
    • Use an OR gate for AAA and BBB.
    • The output of the OR gate will be A+BA + BA+B.
  3. Realize the second part:
    • Use a NOT gate for CCC to get C‾\overline{C}C.
    • Use an OR gate for C‾\overline{C}C and D.
    • The output of this OR gate will be C‾+D\overline{C} + DC+D.
  4. Final Step:
    • Use an AND gate to combine (A+B)(A + B)(A+B) and (C‾+D)(\overline{C} + D)(C+D).
    • The final output is F=(A+B)â‹…(C‾+D)F = (A + B) \cdot (\overline{C} + D)F=(A+B)â‹…(C+D).

Logic Gate Diagram:

  • Inputs: A, B, C, and D
  • Gates:
    • Two OR gates: One for A+BA + BA+B, another for C‾+D\overline{C} + DC+D.
    • One NOT gate for CCC to produce C‾\overline{C}C.
    • One AND gate for combining the two OR gate outputs.

Circuit Diagram Overview:

  1. OR Gate: Inputs are A and B; output is A+BA + BA+B.
  2. NOT Gate: Input is C; output is C‾\overline{C}C.
  3. OR Gate: Inputs are C‾\overline{C}C and D; output is C‾+D\overline{C} + DC+D.
  4. AND Gate: Inputs are A+BA + BA+B and C‾+D\overline{C} + DC+D; output is the final function FFF.

🧠 Key Points of Realization

  • NOT Gate: Used when a variable needs to be inverted (e.g., A‾\overline{A}A).
  • AND Gate: Used to multiply the variables, representing the product (AND) of the terms.
  • OR Gate: Used to add the variables, representing the sum (OR) of the terms.

🔥 Use of Universal Gates in Realization

NAND and NOR gates are universal gates, which means you can use them to implement any Boolean function. In fact, all the basic gates (AND, OR, NOT) can be made using only NAND or NOR gates.

Example: Realizing an AND Gate using NAND Gates:

  • An AND gate can be built using two NAND gates:
    1. First, pass both inputs (A and B) through a NAND gate. The output is A⋅B‾\overline{A \cdot B}A⋅B.
    2. Then, pass the output of this NAND gate through another NAND gate. The final output will be Aâ‹…BA \cdot BAâ‹…B, which is the AND operation.

Example: Realizing an OR Gate using NOR Gates:

  • An OR gate can be made by using three NOR gates:
    1. First, invert each input (A and B) using two separate NOR gates.
    2. Then, OR the outputs of these NOR gates by passing them through a third NOR gate.

🚀 Summary

  1. SOP Realization:
    • AND the variables first, then OR the results.
    • Use AND gates for AND operations and OR gates for OR operations.
  2. POS Realization:
    • OR the variables first, then AND the results.
    • Use OR gates for OR operations and AND gates for AND operations.
  3. Universal Gates (NAND, NOR) can replace any basic gate, making them useful in optimizing circuit designs.