UPSC 2023 Maths Optional Paper 2 Q7a-ii — Step-by-Step Solution

7.5 marks · Section B

Boolean algebra · Numerical Analysis · asked 15× in 13 yrs · Read the full method →

Question

Express the Boolean function

f(x,y,z)=x+\overline{(\overline{x}\cdot\overline{y}+\overline{x}\cdot z)}+z

in disjunctive normal form (DNF) and construct the truth table for the function.

De Morgan to remove the complement; identity $A\bar B+B=A+B$ to absorb the $y\bar z+z$ term.

\overline{\bar x\bar y+\bar x z}=\overline{\bar x(\bar y+z)}=\overline{\bar x}+\overline{\bar y+z}=x+(y\bar z).

f=x+\bigl[x+y\bar z\bigr]+z=x+y\bar z+z\quad(\text{using }x+x=x).

Using the identity $A\bar B+B=A+B$ (with $A=y,B=z$ ):

y\bar z+z=y+z.

Therefore

\boxed{\;f(x,y,z)=x+y+z.\;}