Question

Express each of the Boolean functions in Exercise 3 using the operator \( \downarrow \).

Short answer

The required results are

1. \(\left( {\left( {\left( {x \downarrow y} \right) \downarrow \left( {x \downarrow y} \right)} \right) \downarrow z} \right)\left( {\left( {\left( {x \downarrow y} \right) \downarrow \left( {x \downarrow y} \right)} \right) \downarrow z} \right)\).
2. \(\left( {\left( {x \downarrow z} \right) \downarrow \left( {x \downarrow z} \right) \downarrow \left( {\left( {x \downarrow z} \right) \downarrow \left( {x \downarrow z} \right)} \right)} \right) \downarrow \left( {y \downarrow y} \right)\).
3. \(x\)
4. \(\left( {x \downarrow x} \right) \downarrow \left( {\left( {y \downarrow y} \right) \downarrow \left( {y \downarrow y} \right)} \right)\).

Step-by-step solution

Definition

The complements of an elements\(\overline 0 = 1\)and\(\overline 1 = 0\).

The Boolean sum + or OR is 1 if either term is 1.

The Boolean product (.) or AND is 1 if both terms are 1.

The NAND operator | is 1 if either term is 0.

The NOR operator \( \downarrow \)is 1 if both terms are 0.

Find the solution for part (a).

a)

Use the following results to express the Boolean functions using the operator \( \downarrow \).

1. \(\overline x = x \downarrow x\)
2. \(xy = \left( {x \downarrow x} \right) \downarrow \left( {y \downarrow y} \right)\)
3. \(x + y = \left( {x \downarrow y} \right) \downarrow \left( {x \downarrow y} \right)\)

Consider the function, \(F\left( {x,y,z} \right) = x + y + z\).

Express it using \( \downarrow \)and by using the above results as follows:

\(\begin{aligned}{c}F\left( {x,y,z} \right) &= x + y + z\\& = \left( {x + y} \right) + z\\ &= \left( {\left( {x + y} \right) \downarrow z} \right)\left( {\left( {x + y} \right) \downarrow z} \right)\\ &= \left( {\left( {\left( {x \downarrow y} \right) \downarrow \left( {x \downarrow y} \right)} \right) \downarrow z} \right)\left( {\left( {\left( {x \downarrow y} \right) \downarrow \left( {x \downarrow y} \right)} \right) \downarrow z} \right)\end{aligned}\)

Determine the result of part (b).

b)

Consider the function, \(F\left( {x,y,z} \right) = \left( {x + z} \right)y\)

Express it using \( \downarrow \)and by using the above results as follows:

\(\begin{aligned}{c}F\left( {x,y,z} \right) &= \left( {x + z} \right)y\\ &= \left( {x + z} \right) \downarrow \left( {x + z} \right) \downarrow \left( {y \downarrow y} \right)\\ &= \left( {\left( {x \downarrow z} \right) \downarrow \left( {x \downarrow z} \right) \downarrow \left( {\left( {x \downarrow z} \right) \downarrow \left( {x \downarrow z} \right)} \right)} \right) \downarrow \left( {y \downarrow y} \right)\end{aligned}\)

Evaluate the result of part (c).

c)

Consider the function, \(F\left( {x,y,z} \right) = x\).

Here, \(x\) contains no operators and thus \(x\) is represented as \(x\) itself.

Find the solution of part (d).

(d)

Consider the function, \(F\left( {x,y,z} \right) = x\overline y \).

Express it using \( \downarrow \)and by using the above results as follows:

\(\begin{aligned}{c}F\left( {x,y,z} \right)& = x\overline y \\ &= \left( {x \downarrow x} \right) \downarrow \left( {\overline y \downarrow \overline y } \right)\\ &= \left( {x \downarrow x} \right) \downarrow \left( {\left( {y \downarrow y} \right) \downarrow \left( {y \downarrow y} \right)} \right)\end{aligned}\)

This is the require result.