In some Boolean functions there are some input terms that will never occur, for example in BCD code there are sex invalid combinations 1010,1011,1100, 1110, Since these combinations will never occur, they can be treated ads don ’ t-care condition and can be used to simplify the Boolean function. For example the following table describes a logic function that has an output 1 only when its input is 7,8,9. The Don’t-care condition

Inputs Output ABCDF X 1011X 1100X 1101X 1110X 1111X XXXX 11XX The don’t-care conditions ( denoted by X in the table are combined with the 1’s to simplify the given function AB CD

The resulting POS expression is

The following figure: The resulting POS expression is

Boolean functions with five variables can be simplified using 32- cell Karnaugh map. Two 4-variable maps are used to construct a 5-varibal map. All we need is the cell adjacencies between the two 4-variable map and how to group those adjacent 1’s. Each map contains 16 cells for all combinations of variables ( e.g. B, C, D, E). One map for the remaining variable (e.g. A=0) and the other one for A=1. Five-Variable Karnaugh maps

Example: Minimize the following SOP 5-variable expression: Solution : combining the terms yields to: Five-Variable Karnaugh maps A= A=1

1. The associative law for addition is normally written as a. A + B = B + A b. (A + B) + C = A + (B + C) c. AB = BA d. A + AB = A Ans. b © 2008 Pearson Education

2. The Boolean equation AB + AC = A(B+ C) illustrates a. the distribution law b. the commutative law c. the associative law d. DeMorgan’s theorem Ans. c © 2008 Pearson Education

3. The Boolean expression A. 1 is equal to a. A b. B c. 0 d. 1 Ans. a © 2008 Pearson Education

4. The Boolean expression A + 1 is equal to a. A b. B c. 0 d. 1 Ans. d © 2008 Pearson Education

5. The Boolean equation AB + AC = A(B+ C) illustrates a. the distribution law b. the commutative law c. the associative law d. DeMorgan’s theorem Ans. a © 2008 Pearson Education

6. A Boolean expression that is in standard SOP form is a. the minimum logic expression b. contains only one product term c. has every variable in the domain in every term d. none of the above Ans. c © 2008 Pearson Education

7. Adjacent cells on a Karnaugh map differ from each other by a. one variable b. two variables c. three variables d. answer depends on the size of the map Ans. a

© 2008 Pearson Education 8. The minimum expression that can be read from the Karnaugh map shown is a. X = A b. X = A c. X = B d. X = B Ans. a

© 2008 Pearson Education 9. The minimum expression that can be read from the Karnaugh map shown is a. X = A b. X = A c. X = B d. X = B Ans. d