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Chapter 12: Q7E (page 828)

Design a circuit that implements majority voting for five individuals.

Short Answer

Expert verified

The result is \(abc + abd + abe + acd + ace + ade + bcd + bcd + bde + cde\).

Step by step solution

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01

Defining of gates.

There are three types of gates.

It is also called NOT gate.

02

Construct a circuit.

A circuit that implements five inputs requires. Let the inputs are a, b, c, d ,e.

The outputs will need to be 1 when at least three of the inputs is a 1.Thus the outputs will be 1 when abc, abd ,abe, acd, ace, ade, bcd,bce, bde, or cde is equal to 1. Or equivalently when \(abc + abd + abe + acd + ace + ade + bcd + bcd + bde + cde\) I equal to 1

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Most popular questions from this chapter

Show that \({\bf{x\bar y + y\bar z + \bar xz = \bar xy + \bar yz + x\bar z}}\).

Let \({\bf{x}}\) and \({\bf{y}}\) belong to \(\left\{ {{\bf{0,1}}} \right\}\). Does it necessarily follow that \({\bf{x = y}}\) if there exists a value \({\bf{z}}\) in \(\left\{ {{\bf{0,1}}} \right\}\) such that,

\(\begin{array}{l}{\bf{a) xz = yz?}}\\{\bf{b) x + z = y + z?}}\\{\bf{c) x}} \oplus {\bf{z = y}} \oplus {\bf{z?}}\\{\bf{d) x}} \downarrow {\bf{z = y}} \downarrow {\bf{z?}}\\{\bf{e) x}}|{\bf{z = y}}|z{\bf{?}}\end{array}\)

A Boolean function \({\bf{F}}\) is called self-dual if and only if \({\bf{F}}\left( {{{\bf{x}}_{\bf{1}}}{\bf{, \ldots ,}}{{\bf{x}}_{\bf{n}}}} \right){\bf{ = }}\overline {{\bf{F}}\left( {{{{\bf{\bar x}}}_{\bf{1}}}{\bf{, \ldots ,}}{{{\bf{\bar x}}}_{\bf{n}}}} \right)} \).

Show that \({\bf{F}}\left( {{\bf{x, y, z}}} \right){\bf{ = x y + x z + y z}}\) has the value \(1\) if and only if at least two of the variables \({\bf{x, y}}\), and \({\bf{z}}\) have the value \(1\) .

Use a \({\bf{K}}\)-map to find a minimal expansion as a Boolean sum of Boolean products of each of these functions in the variables \({\bf{w, x, y}}\) and \({\bf{z}}\).

\(\begin{array}{l}{\bf{a) wxyz + wx\bar yz + wx\bar y\bar z + w\bar xy\bar z + w\bar x\bar yz}}\\{\bf{b) wxy\bar z + wx\bar yz + w\bar xyz + \bar wx\bar yz + \bar w\bar xy\bar z + \bar w\bar x\bar yz}}\\{\bf{c) wxyz + wxy\bar z + wx\bar yz + w\bar x\bar yz + w\bar x\bar y\bar z + \bar wx\bar yz + \bar w\bar xy\bar z + \bar w\bar x\bar yz}}\\{\bf{d) wxyz + wxy\bar z + wx\bar yz + w\bar xyz + w\bar xy\bar z + \bar wxyz + \bar w\bar xyz + \bar w\bar xy\bar z + \bar w\bar x\bar yz}}\end{array}\)

Draw the Hasse diagram for the poset consisting of the set of the \({\bf{16}}\)Boolean functions of degree two (shown in Table \({\bf{3}}\) of Section \({\bf{12}}{\bf{.1}}\)) with the partial ordering \( \le \).
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