Math Problem Statement

Minimize the Boolean function F(A, B, C) = Σm(0, 1, 3, 5, 6) using a 3-variable K-map.

Solution

The problem asks to minimize a Boolean function F(A,B,C)F(A, B, C) using a 3-variable Karnaugh map (K-map). The function is given as a sum of minterms:

F(A,B,C)=Σm(0,1,3,5,6)F(A, B, C) = \Sigma m(0, 1, 3, 5, 6)

Step-by-step K-map minimization:

  1. Set up the K-map:

    • The three variables AA, BB, and CC form a 3-variable K-map with 8 cells corresponding to the possible combinations of these variables. These are the minterms from 0 to 7.
    AB \ BC00011110
    000132
    014576
  2. Plot the minterms: From the function F(A,B,C)=Σm(0,1,3,5,6)F(A, B, C) = \Sigma m(0, 1, 3, 5, 6), place 1s in the K-map for the corresponding minterms:

    • Minterm 0: ABCA'B'C'
    • Minterm 1: ABCA'B'C
    • Minterm 3: ABCA'BC
    • Minterm 5: ABCAB'C
    • Minterm 6: ABCABC'

    This results in the following K-map:

    AB \ BC00011110
    001110
    010101
  3. Group the 1s:

    • We can form two groups:
      • A group of 4 covering cells (0, 1, 3, 5): This simplifies to BCB'C.
      • A group of 2 covering cells (1, 3): This simplifies to ACA'C.
      • A group of 2 covering cells (5, 6): This simplifies to ABAB'.
  4. Simplified expression: The minimized Boolean expression is:

    F(A,B,C)=BC+AC+ABF(A, B, C) = B'C + A'C + AB'

Answer:

The correct minimized expression corresponds to Option 3: AB+BC+AC+ABCAB + BC' + AC + A'B'C.

Would you like further details or have any questions? Here are some follow-up questions that you can consider:

  1. How does the K-map help in minimizing Boolean expressions?
  2. What are prime implicants and how do they apply in this case?
  3. Can the minimized expression be verified using Boolean algebra?
  4. What would the expression look like if additional minterms were added?
  5. How does grouping in a K-map improve simplification efficiency?

Tip: Always aim to form the largest possible groups of 1s in a K-map to achieve maximum simplification.

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Math Problem Analysis

Mathematical Concepts

Boolean Algebra
Karnaugh Maps (K-map)
Logic Simplification

Formulas

F(A, B, C) = Σm(0, 1, 3, 5, 6)
Minimized Boolean Expression

Theorems

Karnaugh Map Theorem
Grouping of Minterms

Suitable Grade Level

University-level digital logic or computer science students