Convert F(A, B, C) = (A + $\bar{B}$) ($\bar{B}$ + C) into canonical Product of Sum form.
(A + B + C) (A + B + $ar{C}$) (A + $ar{B}$ + C)
(A + B + C) ($ar{A}$ + B + $ar{C}$) ($ar{A}$ + $ar{B}$ + $ar{C}$)
(A + $ar{B}$ + C) (A + $ar{B}$ + $ar{C}$) ($ar{A}$ + B + $ar{C}$)
(A + B + $ar{C}$) ($ar{A}$ + $ar{B}$ + $ar{C}$) (A + $ar{B}$ + C)
Answer is Right!
Answer is Wrong!
This question was previously asked in
UPSC CISF-AC-EXE – 2020
The given function is $F(A, B, C) = (A + \bar{B}) (\bar{B} + C)$.
We can find the minterms where F is 0 by finding where $(A + \bar{B})=0$ OR $(\bar{B} + C)=0$.
$(A + \bar{B}) = 0$ if and only if $A=0$ AND $\bar{B}=0$, which means $A=0$ and $B=1$. For three variables, this corresponds to minterms 010 ($m_2$) and 011 ($m_3$).
$(\bar{B} + C) = 0$ if and only if $\bar{B}=0$ AND $C=0$, which means $B=1$ and $C=0$. For three variables, this corresponds to minterms 010 ($m_2$) and 110 ($m_6$).
So, F=0 for the union of these minterms: $\{m_2, m_3\} \cup \{m_2, m_6\} = \{m_2, m_3, m_6\}$.
The canonical POS form is the product of the corresponding maxterms $M_2, M_3, M_6$.
The maxterm $M_i$ corresponds to the binary representation of $i$, where a 0 corresponds to the uncomplemented variable and a 1 corresponds to the complemented variable in the sum term.
$M_2$ from 010: $(A + \bar{B} + C)$
$M_3$ from 011: $(A + \bar{B} + \bar{C})$
$M_6$ from 110: $(\bar{A} + \bar{B} + C)$
The canonical POS form is $(A + \bar{B} + C)(A + \bar{B} + \bar{C})(\bar{A} + \bar{B} + C)$.
Comparing this derived form to the options, Option C is $(A + \bar{B} + C) (A + \bar{B} + \bar{C}) (\bar{A} + B + \bar{C})$.
Option C has the first two terms correct (M2 and M3). However, the third term in Option C is $(\bar{A} + B + \bar{C})$, which is $M_5$ (from 101). The correct third term should be $(\bar{A} + \bar{B} + C)$, which is $M_6$ (from 110).
There appears to be an error in the provided options as none exactly matches the derived canonical POS form. However, option C contains two out of the three correct maxterms and is the closest match structurally. Assuming a likely typo in the third term of Option C, it is the most probable intended answer.
– Canonical POS is a product of maxterms.
– Maxterms correspond to the minterms where the function is 0.
– For a variable in a maxterm, it is uncomplemented if its value is 0 in the corresponding minterm binary representation, and complemented if its value is 1.