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Expert-verified Found in: Page 841 ### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095 # Use a $$K{\bf{ - }}$$map to find a minimal expansion as a Boolean sum of Boolean products of each of these functions of the Boolean variables $${\bf{x}}$$ and $${\bf{y}}$$.$$\begin{array}{l}{\bf{a)\bar xy + \bar x\bar y}}\\{\bf{b)xy + x\bar y}}\\{\bf{c)xy + x\bar y + \bar xy + \bar x\bar y}}\end{array}$$

$$(a)$$ The minimum expansion for the sum-of-product is $$\bar x$$

$$(b)$$ The minimum expansion for the sum-of-product is $$x$$

$$(c)$$ The minimum expansion for the sum-of-product is $$1$$

See the step by step solution

## Step 1:Definition

To reduce the number of terms in a Boolean expression representing a circuit, it is necessary to find terms to combine. There is a graphical method, called a Karnaugh map or K-map, for finding terms to combine for Boolean functions involving a relatively small number of variables. You will first illustrate how K-maps are used to simplify expansions of Boolean functions in two variables. You will continue by showing how K-maps can be used to minimize Boolean functions in three variables and then in four variables. Then you will describe the concepts that can be used to extend K-maps to minimize Boolean functions in more than four variables.

$$K{\bf{ - }}$$map for a function in two variables is basically a table with two columns $${\bf{y}}$$ and $${\bf{\bar y}}$$and two rows $${\bf{x}}$$ and $${\bf{\bar x}}$$.

## Step 2: Placing the values in the cell

Place a $$1$$ in the cell corresponding to each term in the given sum $${\bf{\bar xy + \bar x\bar y}}$$.

$${\bf{\bar xy}}$$: place a $$1$$ in the cell corresponding to row $${\bf{\bar x}}$$ and column $${\bf{y}}$$

$${\bf{\bar x\bar y}}$$: place a $$1$$ in the cell corresponding to row $${\bf{\bar x}}$$ and column $${\bf{\bar y}}$$ We note that both ones occur in the row of $${\bf{\bar x}}$$, which means that the minimum expansion for the sum-of-product is $${\bf{\bar x}}$$.

## Step 3: Placing the values in the cell

$$K{\bf{ - }}$$map for a function in two variables is basically a table with two columns $${\bf{y}}$$ and $${\bf{\bar y}}$$and two rows $${\bf{x}}$$ and $${\bf{\bar x}}$$.

Place a $$1$$ in the cell corresponding to each term in the given sum $${\bf{xy + x\bar y}}$$.

$${\bf{xy}}$$: place a $$1$$ in the cell corresponding to row $${\bf{x}}$$ and column $${\bf{y}}$$

$${\bf{x\bar y}}$$: place a $$1$$ in the cell corresponding to row $${\bf{x}}$$ and column $${\bf{\bar y}}$$ We note that both ones occur in the row of $${\bf{x}}$$, which means that the minimum expansion for the sum-of-product is $${\bf{x}}$$.

## Step 4: Placing the values in the cell

$$K{\bf{ - }}$$map for a function in two variables is basically a table with two columns $${\bf{y}}$$ and $${\bf{\bar y}}$$and two rows $${\bf{x}}$$ and $${\bf{\bar x}}$$.

Place a $$1$$ in the cell corresponding to each term in the given sum $${\bf{xy + x\bar y + \bar xy + \bar x\bar y}}$$.

$${\bf{xy}}$$: place a $$1$$ in the cell corresponding to row $${\bf{x}}$$ and column $${\bf{y}}$$

$${\bf{x\bar y}}$$: place a $$1$$ in the cell corresponding to row $${\bf{x}}$$ and column $${\bf{\bar y}}$$

$${\bf{\bar xy}}$$: place a $$1$$ in the cell corresponding to row $${\bf{\bar x}}$$ and column $${\bf{y}}$$

$${\bf{\bar x\bar y}}$$: place a $$1$$ in the cell corresponding to row $${\bf{\bar x}}$$ and column $${\bf{\bar y}}$$ One notes that there is a $$1$$ in each cell of the $$K{\bf{ - }}$$map, which means that the Boolean function will give a $$1$$ for all possible values of $${\bf{x}}$$ and $${\bf{y}}$$. This then implies that the minimum expansion for the sum-of-product is $$1$$. ### Want to see more solutions like these? 