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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 $$K{\bf{ - }}$$maps to find simpler circuits with the same output as each of the circuits shown.a) b) c) a) The output of the given K-map is b) The output of the given K-map is c) Use the zero-property and the output is0.

See the step by step solution

## Step 1: Definition

Recall the following laws:

Idempotent law:

$$\begin{array}{c}{\bf{x + x = x}}\\x \cdot x{\bf{ = x}}\end{array}$$

Commutative law:

$$\begin{array}{c}x{\bf{ + y = y + x}}\\{\bf{xy = yx}}\end{array}$$

Distributive law:

$$\begin{array}{c}{\bf{x + yz = }}\left( {{\bf{x + y}}} \right)\left( {{\bf{x + z}}} \right)\\{\bf{x}}\left( {{\bf{y + z}}} \right){\bf{ = xy + xz}}\end{array}$$

Zero property:

$${\bf{x\bar x = 0}}$$

Domination law

$$\begin{array}{c}{\bf{x + 1 = 1}}\\x \cdot 0 = 0\end{array}$$

Identity law

$$\begin{array}{c}{\bf{x + 0 = x}}\\x \cdot 1{\bf{ = x}}\end{array}$$

## Step 2: AND Gate

A $$K{\bf{ - }}$$map for a function in three variables is a table with four columns $$yz,y\bar z,\bar y\bar z$$ and $$\bar yz$$; which contains all possible combinations of $$y$$ and $$z$$ and two rows $$x$$ and $$\bar x$$.

Place a $$1$$ in the cell corresponding to each term in the given sum $$xyz + \bar xyz$$.

$$xyz$$: place a $$1$$ in the cell corresponding to row $$x$$ and column $$yz$$

$$\bar xyz$$: place a $$1$$ in the cell corresponding to row $$\bar x$$ and column $$yz$$

Note that all $$1$$'s occur in the column $$yz$$, which implies that the simplest Boolean expression is $$yz$$.

Please write all the mathematical equations/expressions/terms of MATHTYPE in BLACK ( Not bold) in the solution part.

Done The Boolean product is represented by an $$AND$$ gate, thus a simpler circuit is then:

Please write all the mathematical equations/expressions/terms of MATHTYPE in BLACK ( Not bold) in the solution part.

Done ## Step 3: Inverter

A $$K -$$map for a function in three variables is a table with four columns $$yz,y\bar z,\bar y\bar z$$ and $$\bar yz$$; which contains all possible combinations of $$y$$ and $$z$$ and two rows $$x$$ and $$\bar x$$.

Place a $$1$$ in the cell corresponding to each term in the given sum $$xy\bar z + x\bar y\bar z + \bar xy\bar z + \bar x\bar y\bar z$$

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

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

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

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

Note that all $$1$$'s occur in the columns $$y\bar z$$ and $$\bar y\bar z$$ (which are all columns containing $$\bar z$$, which implies that the simplest Boolean expression is $$\bar z$$.

Please write all the mathematical equations/expressions/terms of MATHTYPE in BLACK ( Not bold) in the solution part.

CORRECT THROUGHOUT THE FILE

Done The complement is represented by an inverter; thus, a simpler circuit is then: ## Step 4: Inverter and the Boolean product

Simplify the given expression using logical equivalences.Using the commutative, idempotent, distributive, domination law and zero property.

$$\begin{array}{c}\bar xyz\left( {\left( {x + \bar z} \right) + \left( {\bar y + \bar z} \right)} \right) = \bar xyz\left( {x + \bar z + \bar y + \bar z} \right)\\ = \bar xyz\left( {x + \bar y + \bar z + \bar z} \right)\\ = \bar xyz\left( {x + \bar y + \bar z} \right)\\ = x\bar xyz + \bar xy\bar yz + \bar xyz\bar z\end{array}$$

$$\begin{array}{c} = 0 \cdot yz + 0 \cdot \bar xz + 0 \cdot \bar xy\\ = 0 + 0 + 0\\ = 0\end{array}$$

The simplest Boolean expression is $$0$$. However, we cannot express this with a logic gate (if we actually want to use a logic gate), but then use the zero property $$x\bar x = 0$$.The complement is represented by an inverter and the Boolean product by an $$AND$$ gate, thus a simpler circuit is then:

Please write all the mathematical equations/expressions/terms of MATHTYPE in BLACK ( Not bold) in the solution part.

CORRECT THROUGHOUT THE FILE

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