GATE CS Boolean Algebra questions

Consider the following Boolean expression of a function $F$ : $$F(P, Q)=(\bar{P}+Q) \oplus(\bar{P} Q)$$ Which of the following expressions is/are equivalent to $F$ ?

Which one of the following options is not a property of Boolean Algebra? Note: ${+}$ is OR operation, $\cdot$ is AND operation, and ${\prime}$ is NOT operation

Let $X$ be a $3$-variable Boolean function that produces output as $'1'$ when at least two of the input variables are $'1'$. Which of the following statement(s) is/are CORRECT, where $a, b, c, d, e$ are Boolean variables?

Which of the following Boolean algebraic equation(s) is/are CORRECT?

​​​​For a Boolean variable $x$, which of the following statements is/are FALSE?

Consider the following Boolean expression. $F=(X+Y+Z)(\overline X +Y)(\overline Y +Z)$ Which of the following Boolean expressions is/are equivalent to $\overline F$ (complement of $F$)?

Consider a Boolean function $f(w,x,y,z)$ such that $$\begin{array}{lll} f(w,0,0,z) & = & 1 \\ f(1,x,1,z) & =& x+z \\ f(w,1,y,z) & = & wz +y \end{array}$$The number of literals in the minimal sum-of-products expression of $f$ is _________

Consider the Boolean function $z(a,b,c)$. Which one of the following minterm lists represents the circuit given above?

Which one of the following is NOT a valid identity?

Consider three $4$-variable functions $f_1, f_2$, and $f_3$, which are expressed in sum-of-minterms as $f_1=\Sigma(0,2,5,8,14),$ $f_2=\Sigma(2,3,6,8,14,15),$ $f_3=\Sigma (2,7,11,14)$ For the following circuit with one AND gate and one XOR gate the output function $f$ can be expressed as:

What is the minimum number of $2$-input NOR gates required to implement a $4$ -variable function expressed in sum-of-minterms form as $f=\Sigma(0,2,5,7, 8, 10, 13, 15)?$ Assume that all the inputs and their complements are available. Answer: _______

Let $\oplus$ and $\odot$ denote the Exclusive OR and Exclusive NOR operations, respectively. Which one of the following is NOT CORRECT?

If $w, x, y, z$ are Boolean variables, then which one of the following is INCORRECT?

Consider the Boolean operator # with the following properties : $x \# 0 = x, x \# 1=\overline{x}, x \# x = 0$ and $x \# \overline{x} = 1.$ Then $x\#y$ is equivalent to

Let, $x_{1} ⊕ x_{2} ⊕ x_{3} ⊕ x_{4}= 0$ where $x_{1}, x_{2}, x_{3}, x_{4}$ are Boolean variables, and $⊕$ is the XOR operator. Which one of the following must always be TRUE?

Consider the operations $\textit{f (X, Y, Z) = X'YZ + XY' + Y'Z'}$ and $\textit{g (X, Y, Z) = X'YZ + X'YZ' + XY}$ Which one of the following is correct?

The number of min-terms after minimizing the following Boolean expression is _______. $[D'+AB'+A'C+AC'D+A'C'D]'$

Given the function $F = P' +QR$, where $F$ is a function in three Boolean variables $P, Q$ and $R$ and $P'=!P$, consider the following statements. $(S1) F = \sum(4, 5, 6)$ $(S2) F = \sum(0, 1, 2, 3, 7)$ $(S3) F = \Pi (4, 5, 6)$ $(S4) F = \Pi (0, 1, 2, 3, 7)$ Which of the following is true?

The total number of prime implicants of the function $f(w, x, y, z) = \sum (0, 2, 4, 5, 6, 10)$ is __________

The dual of a Boolean function $F(x_1,x_2,\dots,x_n,+, .,')$, written as $F^D$ is the same expression as that of $F$ with $+$ and $⋅$ swapped. $F$ is said to be self-dual if $F = F^D$. The number of self-dual functions with $n$ Boolean variables is