# Boolean Algebra

 Question 1
Select the Boolean function(s) equivalent to $x+yz$, where $x,y$, and $z$ are Boolean variables, and + denotes logical OR operation.
 A x+z+xy B (x+y)(x+z) C x+xy+yz D x+xz+xy
GATE EC 2022   Digital Circuits
Question 1 Explanation:
A. x + z + xy = x(1 + y) + z = x + z
B. (x + y) (x + z) = x + xz + xy + yz = x(1 + y + z) + yz = x + yz
C. x + xy + yz = x(1+y) + yz = x + yz
D. x + xz + xy = x (1 + z + y) = x
 Question 2
A function F(A,B,C) defined by three Boolean variables A, B and C when expressed as sum of products is given by

$F=\bar{A}\cdot \bar{B} \cdot \bar{C} + \bar{A}\cdot B \cdot \bar{C} + A\cdot \bar{B} \cdot \bar{C}$

where,$\bar{A},\bar{B} \;and \; \bar{C}$ are complements of the respective variable. The product of sums (POS) form of the function F is
 A $F=(A+B+C)\cdot (A+\tilde{B}+C)\cdot (\bar{A}+B+C)$ B $F=(\bar{A}+\bar{B}+\bar{C})\cdot (\bar{A}+B+\bar{C})\cdot (A+\bar{B}+\bar{C})$ C $F=(A + B + \bar{C}) \cdot (A + \bar{B} + \bar{C} ) \cdot (\bar{A} + B + \bar{C}) \cdot$ $(\bar{A}+\bar{B}+C) \cdot (\bar{A}+\bar{B}+\bar{C})$ D $F=(\bar{A} + \bar{B} + C) \cdot (\bar{A} + B + C) \cdot$ $(A + B + \bar{C}) \cdot (A+B+C)$
GATE EC 2018   Digital Circuits
Question 2 Explanation:
\begin{aligned} F(A, B, C, D) &=\bar{A} \bar{B} \bar{C}+\bar{A} B \bar{C}+A \bar{B} \bar{C} \\ &=\Sigma m(0,2,4)=\Pi M(1,3,5,6,7) \\ =&(A+B+\bar{C})(A+\bar{B}+\bar{C})(\bar{A}+B+\bar{C}) \\ &(\bar{A}+\bar{B}+C)(\bar{A}+\bar{B}+\bar{C}) & \end{aligned}

 Question 3
Which one of the following gives the simplified sum of products expression for the Boolean function $F=m_{0}+m_{2}+m_{3}+m_{5}$, where $m_{0},m_{2},m_{3},m_{5}$, are minterms corresponding to the inputs A, B and C with A as the MSB and C as the LSB?
 A $\bar{A}B+\bar{A}\bar{B}\bar{C}+A\bar{B}C$ B $\bar{A}\bar{C}+\bar{A}B+A\bar{B}C$ C $\bar{A}\bar{C}+A\bar{B}+A\bar{B}C$ D $\bar{A}BC+\bar{A}\bar{C}+A\bar{B}C$
GATE EC 2017-SET-1   Digital Circuits
Question 3 Explanation:
Given Boolean function is,
$F=m_{0}+m_{2}+m_{3}+m_{5}$
It can be minimized by using K-map as given below.

$F=\bar{A} \bar{C}+\bar{A} B+A \bar{B} C$
 Question 4
Following is the K-map of a Boolean function of five variables P, Q, R, S and X. The minimum sum-of-product (SOP) expression for the function is
 A $\bar{P}\bar{Q}S\bar{X}+P\bar{Q}S\bar{X}+Q\bar{R}\bar{S}X+QR\bar{S}X$ B $\bar{Q}S\bar{X}+Q\bar{S}X$ C $\bar{Q}SX+Q\bar{S}\bar{X}$ D $\bar{Q}S+Q\bar{S}$
GATE EC 2016-SET-3   Digital Circuits
Question 4 Explanation:

$\therefore$ minimum sum of product expression of the function is
$=\bar{Q}S\bar{X}+Q\bar{S}X$
 Question 5
A function of Boolean variables X, Y and Z is expressed in terms of the min-terms as
$F(X,Y,Z)=\sum (1,2,5,6,7)$
Which one of the product of sums given below is equal to the function F(X,Y,Z)?
 A $(\bar{X}+\bar{Y}+\bar{Z})\cdot (\bar{X}+Y+Z)\cdot (X+\bar{Y}+\bar{Z})$ B $(X+Y+Z)\cdot (X+ \bar{Y}+\bar{Z})\cdot (\bar{X}+Y+Z)$ C $(\bar{X}+\bar{Y}+Z)\cdot (\bar{X}+Y+\bar{Z})\cdot (X+\bar{Y}+Z)$ $\cdot (X+Y+\bar{Z})\cdot (X+Y+Z)$ D $(X+Y+\bar{Z})\cdot (\bar{X}+Y+Z)\cdot (\bar{X}+Y+\bar{Z})$ $\cdot (\bar{X}+\bar{Y}+Z)\cdot (\bar{X}+\bar{Y}+\bar{Z})$
GATE EC 2015-SET-2   Digital Circuits
Question 5 Explanation:
\begin{aligned} F(X, Y, Z) &=\Sigma(1,2,5,6,7) \\ &=\pi[0,3,4] \\ =(X+Y+Z)&(X+\bar{Y}+\bar{Z})(\bar{X}+Y+Z) \end{aligned}

There are 5 questions to complete.