Question

Find the sum-of-products expansions of the Boolean function \({\bf{F}}\left( {{\bf{x, y, z}}} \right)\) that equals 1 if and only if

a) \({\bf{x = 0}}\)

b) \({\bf{xy = 0}}\)

c) \({\bf{x + y = 0}}\)

d) \({\bf{xyz = 0}}\)

Short answer

The sum of products expansions are

1. The sum of product is \(F\left( {x,y,z} \right) = \overline x \overline y \overline z + \overline x \overline y z + \overline x y\overline z + \overline x yz\).
2. The sum of product is \(F\left( {x,y,z} \right) = \overline x \overline y \overline z + \overline x \overline y z + \overline x y\overline z + \overline x yz + x\overline y \overline z + x\overline y z\).
3. The sum of product is \(F\left( {x,y,z} \right) = \overline x \overline y \overline z + \overline x \overline y z\).
4. The sum of product is \(F\left( {x,y,z} \right) = xyz\).

Step-by-step solution

Definition

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

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

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

(a) Find the result.

Here \(F\left( {x,y,z} \right) = 1\) if and only if \(x = 0\).

By getting the values of F get the sum of products form.

X	Y	Z	\(F\left( {x,y,z} \right)\)	Sum of products terms
0	0	0	1	\(\overline x \overline y \overline z \)
0	0	1	1	\(\overline x \overline y z\)
0	1	0	1	\(\overline x y\overline z \)
0	1	1	1	\(\overline x yz\)
1	0	0	0	\(x\overline y \overline z \)
1	0	1	0	\(x\overline y z\)
1	1	0	0	\(xy\overline z \)
1	1	1	0	\(xyz\)

The sum of product expansion of G then contains a term for every row that has 1 in the last column. A term is the product of the three variables.

\(F\left( {x,y,z} \right) = \overline x \overline y \overline z + \overline x \overline y z + \overline x y\overline z + \overline x yz\).

(b) Evaluate for result.

Here \(F\left( {x,y,z} \right) = 1\) if and only if \(xy = 0\).

By getting the values of F get the sum of products form.

X	Y	Z	\(xy\)	\(F\left( {x,y,z} \right)\)	Sum of products terms
0	0	0	0	1	\(\overline x \overline y \overline z \)
0	0	1	0	1	\(\overline x \overline y z\)
0	1	0	0	1	\(\overline x y\overline z \)
0	1	1	0	1	\(\overline x yz\)
1	0	0	0	1	\(x\overline y \overline z \)
1	0	1	0	1	\(x\overline y z\)
1	1	0	1	0	\(xy\overline z \)
1	1	1	1	0	\(xyz\)

The sum of product expansion of G then contains a term for every row that has 1 in the last column .A term is the product of the three variables.

\(F\left( {x,y,z} \right) = \overline x \overline y \overline z + \overline x \overline y z + \overline x y\overline z + \overline x yz + x\overline y \overline z + x\overline y z\).

(c) Determine the sum of product.

Here \(F\left( {x,y,z} \right) = 1\) if and only if \(x + y = 0\).

By getting the values of F get the sum of products form.

X	Y	Z	\(x + y\)	\(F\left( {x,y,z} \right)\)	Sum of products terms
0	0	0	0	1	\(\overline x \overline y \overline z \)
0	0	1	0	1	\(\overline x \overline y z\)
0	1	0	1	0	\(\overline x y\overline z \)
0	1	1	1	0	\(\overline x yz\)
1	0	0	1	0	\(x\overline y \overline z \)
1	0	1	1	0	\(x\overline y z\)
1	1	0	1	0	\(xy\overline z \)
1	1	1	1	0	\(xyz\)

The sum of product expansion of G then contains a term for every row that has 1 in the last column .A term is the product of the three variables.

\(F\left( {x,y,z} \right) = \overline x \overline y \overline z + \overline x \overline y z\)

(d) Determine the sum of product.

Here \(F\left( {x,y,z} \right) = 1\) if and only if \(xyz = 0\).

By getting the values of F get the sum of products form.

X	Y	Z	\(xyz\)	\(F\left( {x,y,z} \right)\)	Sum of products terms
0	0	0	0	0	\(\overline x \overline y \overline z \)
0	0	1	0	0	\(\overline x \overline y z\)
0	1	0	0	0	\(\overline x y\overline z \)
0	1	1	0	0	\(\overline x yz\)
1	0	0	0	0	\(x\overline y \overline z \)
1	0	1	0	0	\(x\overline y z\)
1	1	0	0	0	\(xy\overline z \)
1	1	1	1	0	\(xyz\)

The sum of product expansion of G then contains a term for every row that has 1 in the last column .A term is the product of the three variables.

\(F\left( {x,y,z} \right) = xyz\).

Therefore, these are the sum of product expansion of Boolean functions.