Minterms, Maps, and Simplification

Tom Kelliher, CS 240

Jan. 30, 2012



How to order the rows of a truth table: 0 at the top; $2^n - 1$ at the bottom. Example: two-input AND.


Read 2.5.

Written assignment: Some of the Boolean manipulation problems are tricky -- start early.

From Last Time

Logic gates and Boolean algebra.


  1. Minterms and products

  2. Simplification using Karnaugh maps.

Coming Up

Karnaugh map manipulation; don't cares.

Minterms and Products

  1. What is a product? A sum?

  2. Definition of a minterm: A product term containing all literals, complemented or not complemented.

    Examples in three variables ($X$, $Y$, $Z$). Identify which are minterms and which are not: $XYZ$, $X\overline{Y}Z$, $Z$, $XZ$.

  3. Sum of minterms. Can be derived directly from a truth table.

    Example: sum output of a full binary adder. Derive truth table and sum of minterms equation. Observe $F(a, b, c_i) = \sum m(1, 2, 4, 7)$ and relationship to even parity (exclusive or).

  4. Product of sums form and difference from sum of minterms (products).

Karnaugh Maps

  1. A graphical tool for minimizing sum of minterm expressions.

  2. Two-variable maps:
    1. Structure; literal and value labels.

    2. Theory: Show simplification of $F(A, B) = \sum m(0, 1)$ given
        0 1
      $A$ 0 $\overline{A}~\overline{B}$ $\overline{A}B$
        1 $A\overline{B}$ $AB$
      Actual Karnaugh map:
        0 1
      $A$ 0 1 1
        1 0 0

  3. Three-Variable maps:
    1. Structure and connectivity.

    2. Examples: Sum and carry-out of full binary adder.

  4. Four-Variable maps:
    1. Structure and connectivity.

    2. Example: Product bit 1 of two-bit multiplier. (Start with ``product'' table and then produce truth table for bit 1.)

  5. Five-Variable maps? Higher?

Thomas P. Kelliher 2012-01-27
Tom Kelliher