Question

select the boolean expression that is equivalent to the function defined in the table below:

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

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

Explanation:

Step1: Recall min - term concept

In a truth - table, each row where the output function \(f(x,y,z)\) is 1 corresponds to a min - term. A min - term for a 3 - variable function \(x,y,z\) is of the form \(\overline{x}^a\overline{y}^b\overline{z}^c\), where \(a,b,c\in\{0,1\}\) and \(\overline{x}^0 = x,\overline{x}^1=\overline{x}\).

Step2: Identify min - terms from table

When \(x = 0,y = 1,z = 0\), the min - term is \(\overline{x}y\overline{z}\). When \(x = 0,y = 1,z = 1\), the min - term is \(\overline{x}yz\). When \(x = 1,y = 0,z = 1\), the min - term is \(x\overline{y}z\).

Step3: Write the sum - of - min - terms expression

The Boolean function \(f(x,y,z)\) in sum - of - min - terms form is the sum of these min - terms for which \(f(x,y,z)=1\), so \(f(x,y,z)=\overline{x}y\overline{z}+\overline{x}yz + x\overline{y}z\).

Answer:

\(\overline{x}y\overline{z}+\overline{x}yz + x\overline{y}z\) (corresponding to the first option in the multiple - choice list)