Lecture 11: Prime Implicants and Minimal Sums

Key Topics

• Identification of Prime Implicants
• Essential Prime Implicants
• Minimal Sums

Prime Implicants in Carnot Maps

• Prime Implicants Found:
  • T'
  • R'
• Both T' and R' are of size 4.
• No other prime implicants of size 4 exist.

Groupings

• Size 2 Groupings:
  • Present but entirely contained within previously identified prime implicants.
• Groupings of 1:
  • Also contained within prime implicants.

Identifying Essential Prime Implicants

• Essential One Cells:
  • Required to determine essential prime implicants.
  • Essential one cells identified at specific positions, except the upper left and right corners.
• Essential Prime Implicants:
  • T' is essential due to specific essential one cells.
  • R' is essential for the same reason.

Minimal Sums

• Function F3 of R, S, and T:
  • Minimal sum includes essential prime implicants T' and R'.
• Expression Reduction:
  • From 6 literals to 2 literals using Carnot maps.

Methods of Verification

• Truth Table Comparison:
  • Confirm equivalence by comparing truth tables of each expression.
• Algebraic Methods:
  • Show equivalence algebraically.

Problems for Lecture 11

Problem 11.1

• Task: Find all minimal sums of ( f(x, y, z) = \sum m(1, 3, 5, 7) ).
• Choices: Four options provided.

Problem 11.2

• Task: Find all minimal sums of ( g(x, y, z) = \prod M(2, 4) ).
• Choices: Four options provided.

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Lecture 11 concluded with a discussion of problems and methods to find minimal sums using provided techniques. Good luck with your studies!