Question

Show that

(a) \(\overline {\bf{x}} {\bf{ = x}} \downarrow {\bf{x}}\)

(b) \({\bf{xy = (x}} \downarrow {\bf{x)}} \downarrow {\bf{(y}} \downarrow {\bf{y)}}\)

(c) \({\bf{x + y = (x}} \downarrow {\bf{y)}} \downarrow {\bf{(x}} \downarrow {\bf{y)}}\)

Short answer

By using the NOR operator get the results.

(a) \(\overline x = x \downarrow x\)

(b) \(xy = \left( {x \downarrow x} \right) \downarrow \left( {y \downarrow y} \right)\)

(c) \(x + y = \left( {x \downarrow y} \right) \downarrow \left( {x \downarrow y} \right)\)

Step-by-step solution

Definition

The complements of 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.

Show the result of \(\overline {\bf{x}} {\bf{ = x}} \downarrow {\bf{x}}\).    (a)

Here \(\overline x = x \downarrow x\)

The NOR operator is 1 if either term is 0. Thus \(x \downarrow x\)is 1 if \(x\) is 0.

x	\(\overline x \)	\(x \downarrow x\)
0	1	1
1	0	0

The last two columns of the table are identical, which implies: \(\overline x = x \downarrow x\)

Evaluate the result of \({\bf{xy = (x}} \downarrow {\bf{x)}} \downarrow {\bf{(y}} \downarrow {\bf{y)}}\).(b)

Here \(xy = \left( {x \downarrow x} \right) \downarrow \left( {y \downarrow y} \right)\)

The NAND operator is 1 if either term is 0. Thus \(x \downarrow x\)is 1 if \(x\) is 0.

X	Y	\(x \downarrow x\)	\(\left( {y \downarrow y} \right)\)	\(xy\)	\(\left( {x \downarrow x} \right) \downarrow \left( {y \downarrow y} \right)\)
0	0	1	1	0	0
0	1	1	0	0	0
1	0	0	1	0	0
1	1	0	0	1	1

The last two columns of the table are identical, which implies: \(xy = \left( {x \downarrow x} \right) \downarrow \left( {y \downarrow y} \right)\)

Determine the result of \({\bf{x + y = (x}} \downarrow {\bf{y)}} \downarrow {\bf{(x}} \downarrow {\bf{y)}}\).(c)

Here \(x + y = \left( {x \downarrow y} \right) \downarrow \left( {x \downarrow y} \right)\)

The NAND operator is 1 if either term is 0. Thus \(x \downarrow y\)is 1 if \(x\) is 0.

X	Y	\(x \downarrow y\)	\(x + y\)	\(\left( {x \downarrow y} \right) \downarrow \left( {x \downarrow y} \right)\)
0	0	1	0	0
0	1	0	1	1
1	0	0	1	1
1	1	0	1	1

The last two columns of the table are identical, which implies: \(x + y = \left( {x \downarrow y} \right) \downarrow \left( {x \downarrow y} \right)\)

Therefore, by using the NOR operator get the results.