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Chapter 12: Problem 5

a) Draw a K-map for a function in three variables. Put a 1 in the cell that represents \(\overline{x} y \overline{z}\) . b) Which minterms are represented by cells adjacent to this cell?

Short Answer

Expert verified
The K-map cell for \(\backslash{overline{x} y \overline{z}}\) is identified and filled in. Adjacent cells represent minterms differing by one variable from 010 (1 1 2 3).

Step by step solution

01

鈥� Understand the K-map Layout for 3 Variables

A Karnaugh Map (K-map) for three variables has 8 cells, each representing a minterm. The variables can be arranged in a 2x4 grid. Label the columns as combinations of two variables (e.g., AB) and the rows as the remaining variable (e.g., C).
02

鈥� Identify the Minterm \(\backslash{overline{x} y \overline{z}}\)

The term \(\backslash{overline{x} y \overline{z}}\) represents a scenario where x is 0, y is 1, and z is 0. Find the corresponding cell for this combination.
03

鈥� Populate the K-map

Place a 1 in the cell representing \(\backslash{overline{x} y \overline{z}}\). This cell can be found by locating where row \(\backslash{overline{z}}\) meets the column \(\backslash{overline{x} y}\).
04

鈥� Determine Adjacent Cells

Adjacent cells in a K-map differ by only one variable. Identify the cells adjacent to \(\backslash{overline{x} y \overline{z}}\). These will represent the minterms requiring only one-bit change.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

minterms
A minterm in Boolean algebra is a specific combination of variables that results in the function output being 1. Each minterm corresponds to a unique combination of values for the variables in a Boolean function.

For example, in a function with three variables (x, y, z), there are 8 possible minterms, one for each combination of the variables. The term \(\backslash{overline{x} y \backslash{overline{z}}}\) is a minterm where x is 0, y is 1, and z is 0.

Minterms are essential in constructing Karnaugh Maps (K-maps) because they help us to visualize combinations that result in a logical 1.
  • Each cell in a K-map represents a different minterm.
  • Populating the K-map involves marking '1' in the cells corresponding to minterms that make the function true.
adjacent cells
Adjacent cells in a Karnaugh Map are cells that differ by only one variable. This adjacency is crucial when simplifying Boolean expressions, as it allows for grouping and reducing terms.

To find adjacent cells in a K-map, look for cells that are next to each other, whether horizontally, vertically, or even wrapping around the edges. Each cell should only differ by one bit from its neighbor.

For example, if you place a 1 in the cell representing \( \backslash{overline{x} y \backslash{overline{z}}} \), its adjacent cells would be those representing:
  • \( \backslash{overline{x} y z} \) (changing \( \backslash{overline{z}} \) to \(z) \)
  • \( \backslash{x} y \backslash{overline{z}} \) (changing \( \backslash{overline{x}} \) to \(x) \)
  • \( \backslash{overline{x} \backslash{overline{y}} \backslash{overline{z}} \) (changing \(y \) to \( \backslash{overline{y}}) \)
three variables
A Karnaugh Map with three variables contains 8 cells because each variable can be either 0 or 1, yielding 23 = 8 combinations.

Arrange the three variables in a 2x4 grid for clarity:
The rows typically represent one variable (e.g., z), while the columns represent combinations of the other two variables (e.g., x and y). We label the cells according to the values of these variables.

For example, the cells may be filled like this:
  • The first column could represent \( \backslash{overline{x} \backslash{overline{y}}} \)
  • The second column could be \( \backslash{overline{x} y \)
  • The third column \(x \backslash{overline{y}}\), and so on.
Knowing how to read and populate this grid allows you to simplify Boolean expressions accurately.

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Most popular questions from this chapter

Draw the 4 -cube \(Q_{4}\) and label each vertex with the minterm in the Boolean variables \(w, x, y,\) and \(z\) associated with the bit string represented by this vertex. For each literal in these variables, indicate which 3 -cube \(Q_{3}\) that is a subgraph of \(Q_{4}\) represents this literal. Indicate which 2 -cube \(Q_{2}\) that is a subgraph of \(Q_{4}\) represents the products \(w z, \overline{x} y,\) and \(\overline{y} \overline{z}\)

A multiplexer is a switching circuit that produces as output one of a set of input bits based on the value of control bits. Construct a multiplexer using AND gates, OR gates, and inverters that has as input the four bits \(x_{0}, x_{1}, x_{2},\) and \(x_{3}\) and the two control bits \(c_{0}\) and \(c_{1} .\) Set up the circuit so that \(x_{i}\) is the output, where \(i\) is the value of the two-bit integer \(\left(c_{1} c_{0}\right)_{2} .\)

Another way to find a Boolean expression that represents a Boolean function is to form a Boolean product of Boolean sums of literals. Exercises \(7-11\) are concerned with representations of this kind. Find a Boolean sum containing either \(x\) or \(\overline{x},\) either \(y\) or \(\overline{y},\) and either \(z\) or \(\overline{z}\) that has the value 0 if and only if $$ \begin{array}{ll}{\text { a) } x=y=1, z=0 .} & {\text { b) } x=y=z=0} \\\ {\text { c) } x=z=0, y=1}\end{array} $$

Show that a complemented, distributive lattice is a Boolean algebra.

Use a K-map to find a minimal expansion as a Boolean sum of Boolean products of each of these functions of the Boolean variables \(x\) and \(y .\) a) \(\overline{x} y+\overline{x} \overline{y}\) b) \(x y+x \overline{y}\) c) \(x y+x \overline{y}+\overline{x} y+\overline{x} \overline{y}\)
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