Question

Consider the following Boolean function: F = xy’z + x’y’z + w’xy + wx’y + wxy

(a) Obtain the truth table for F.
(b) Draw the logic diagram for the original expression for F.
(c) Use Boolean algebra to simplify the function to a minimum number of literals.

Answer 1

Step-by-step explanation:

A.)

You just pick values for x, y and z, then work out F for each case. Since you have four variables, the values for x, y and z in your table will be as follows:

w x y z

0 0 0 0

0 0 0 1

0 0 1 0

0 0 1 1

0 1 0 0

0 1 0 1

0 1 1 0

0 1 1 1

1 0 0 0

1 0 0 1

1 0 1 0

1 0 1 1

1 1 0 0

1 1 0 1

1 1 1 0

1 1 1 1

Now just work out F for each case. It would be helpful to add a few extra coloumns since that's a pretty big equation. Here's how I would do it (notice the titles of the columns. I've worked out the first two for you to help you understand)

w | x | y | z | xy'z | x?y?z | w?xy | wx?y | wxy | F

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0

0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1

B.)

Cheack attachment for table

C.)

F = xy'z + x'y'z + w'xy + wx'y + wxy

F = (x+x')(y'z) + w'xy + (x+x')(wy)

F = y'z + w'xy + wy

F = y'z + y(w'x + w)

F = y'z + y(w + x)

F = y'z + wy + xy

D)

Logic Circuits, Boolean Algebra, and Truth Tables

: Truth Tables

A truth table is a chart of 1s and 0s arranged to indicate the results (or outputs) of all possible inputs. The list of all possible inputs are arranged in columns on the left and the resulting outputs are listed in columns on the right. There are 2 to the power n possible states (or combination of inputs). For example with three inputs there are 2^3=8 possible combination of inputs. (Audio)

E)

: Logic Diagram

A logic diagram uses the pictoral description of logic gates in combination to represent a logic expression. An example below shows a logic diagram with three inputs (A, B, and C) and one output (Y). The interpretation of this will become clear in the following sections.