Question

multiple choice 1 point
select the boolean expression that is equivalent to the function defined in the table blow:

x	y	f(x,y,z)
0	0	0	1
0	0	1	0
0	1	0	1
0	1	1	0
1	0	0	0
1	0	1	0
1	1	0	0
1	1	1	1

\\(\overline{x}y\overline{z}+x\overline{y}z + xyz\\)
\\(\overline{x}y\overline{z}+\overline{x}y\overline{z}+xyz\\)
\\(\overline{x}y\overline{z}+x\overline{y}z\\)
\\(\overline{x}y\overline{z}-+\overline{x}y\overline{z}+xyz\\)
multiple choice 1 point
select the description that characterizes the boolean expression: \\(\overline{x}y\overline{z}\\)
cnf, but not dnf
neither cnf nor dnf
cnf and dnf
dnf, but not cnf

Explanation:

5.

Step1: Recall min - term concept

In a truth - table, when \(f(x,y,z) = 1\), we form min - terms. For \(x = 0,y = 0,z = 0\), the min - term is \(\overline{x}\overline{y}\overline{z}\); for \(x = 0,y = 1,z = 0\), the min - term is \(\overline{x}y\overline{z}\); for \(x = 1,y = 1,z = 1\), the min - term is \(xyz\).

Step2: Write the sum - of - min - terms

The Boolean function \(f(x,y,z)\) in sum - of - min - terms (SOP or DNF) is the sum of the min - terms for which \(f(x,y,z)=1\). So \(f(x,y,z)=\overline{x}\overline{y}\overline{z}+\overline{x}y\overline{z}+xyz\).

A product - of - sums (POS or CNF) is a conjunction of max - terms, and a sum - of - products (SOP or DNF) is a disjunction of min - terms. The expression \(\overline{x}y\overline{z}\) is a single min - term (a product of literals), so it is in DNF. Also, a single min - term can be considered as a special case of CNF where there is only one product term.

Answer:

\(\overline{x}\overline{y}\overline{z}+\overline{x}y\overline{z}+xyz\)

6.