Boolean Algebra · Digital Logic · GATE CSE

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

Which one of the following is NOT a valid identity?

Consider the Boolean operator $$ \ne $$ with the following properties: $$x \ne 0 = x,\,\,x \ne 1 = \overline x ,\,\,x \ne x = 0$$ and $$x \ne \overli...

Let, $${x_1} \oplus {x_2} \oplus {x_3} \oplus {x_4} = 0$$ where $${x_1},\,{x_2},\,{x_3},\,{x_4}$$ are Boolean Variables, and $$ \oplus $$ is the $$XOR...

Let $$ \ne $$ be a binary operator defined as $$X \ne Y = X' + Y'$$ where $$𝑋$$ and $$𝑌$$ are Boolean variables. Consider the following two statemen...

Consider the following combinational function block involving four Boolean variables $$x, y, a,$$ $$b$$ where $$x, a, b$$ are inputs and $$y$$ is the...

The dual of a Boolean function $$F\left( {{x_1},{x_2},\,....,\,{x_n},\, + , \cdot ,'} \right),$$ written as $${F^D}$$, is the same expression as that ...

Consider the following Boolean expression for $$F:$$ $$F\left( {P,\,Q,\,R,\,S} \right) = PQ + \overline P QR + \overline P Q\overline R S.$$ The mini...

Which one of the following expressions does NOT represent exclusive NOR of $$x$$ and $$y?$$

Which one of the following circuits is NOT equivalent to a $$2$$-input $$XNOR$$ (exclusive NOR) gate

The simplified $$SOP$$ (Sum of product) form of the Boolean expression $$\left( {P + \overline Q + \overline R } \right).\left( {P + \overline Q + ...

The minters expansion of $$f\left( {P,Q,R} \right) = PQ + Q\overline R + P\overline R $$ is

What is the minimum number of gates required to implement the Boolean function $$(AB+C)$$ if we have to use only $$2$$-input NOR gates?

Given $${f_1},$$ $${f_3},$$ and $$f$$ in canonical sum of products form (in decimal) for the circuit. $$${f_1} = \sum {m\left( {4,5,6,7,8} \right)} $$...

Consider the following Boolean function with four variables $$F\left( {w,\,x,\,y,\,z} \right) = \sum {\left( {1,\,3,\,4,\,6,\,9,\,11,\,12,\,14} \righ...

The Boolean function $$x'y' +xy +x'y$$ is equivalent to

Let $$^ * $$ be defined as $${x^ * }y = \overline x + y,$$ Let $$z = {x^ * }y.$$ Value of $${z^ * }x$$ is

Consider 4-variable functions $f1, f2, f3, f4$ expressed in sum-of-minterms form as given below.$f1 = \sum(0,2,3,5,7,8,11,13)$$f2 = \sum(1,3,5,7,11,13...

Consider a Boolean expression given by $F(X, Y, Z) = \Sigma(3,5,6,7)$.Which of the following statements is/are CORRECT?

Consider a Boolean function f(w, x, y, z) such that  f(w, 0, 0, z) = 1 f(1, x, 1, z) = x + z f(w, 1, y, z) = wz + y The number of literals in...

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

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

Consider three 4-variable functions f1, f2 and f3, which are expressed in sum-of-minterms as f1 = Σ(0, 2, 5, 8, 14), f2 = Σ(2, 3, 6, 8, 14, 15), f...

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

Consider a carry lookahead adder for adding two $$n$$-bit integers, built using gates of fan-in at most two. The time to perform addition using this a...

Consider the operations $$f\left( {x,y,z} \right) = X'YZ + XY' + Y'Z'$$ and $$g\left( {x,y,z} \right) = X'YZ + X'YZ' + XY$$. Which one of the followi...

The binary operator $$ \ne $$ is defined by the following truth table. .tg {border-collapse:collapse;border-spacing:0;} .tg td{font-family:Arial, sa...

The total number of prime implicants of the function $$f\left( {w,x,y,z} \right) = \sum {\left( {0,2,4,5,6,10} \right)} $$ _________________.

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 stat...

The number of min-terms after minimizing the following Boolean expression is _______________ . $$$\left[ {D' + AB' + A'C + AC'D + A'C'D} \right]'$$$

A half adder is implemented with $$XOR$$ and $$AND$$ gates. A full adder is implemented with two half adders and one $$OR$$ gate. The propagation dela...

Let $$ \oplus $$ denote the exclusive $$OR\left( {XOR} \right)$$ operation. Let $$'1'$$ and $$'0'$$ denote the binary constants. Consider the followin...

What is the Boolean expression for the output f of the combinational logic circuit of NOR gates given below? ...

If $$P, Q, R$$ are Boolean variables, then $$\left( {P + \overline Q } \right)$$ $$\left( {P.\overline Q + P.R} \right)\left( {\overline P .\overline...

Let $$f\left( {w,x,y,z} \right) = \sum {\left( {0,4,5,7,8,9,13,15} \right).} $$ Which of the following expressions are NOT equivalent to $$f?$$ $$(P)...

Consider a Boolean function $$f(w, x, y, z).$$ Suppose that exactly one of its inputs is allowed to change at a time. If the function happens to be tr...

Which are the essential prime implicants of the following Boolean function? $$F\left( {a,b,c} \right) = {a^1}c + a{c^1} + {b^1}c$$

$$f\left( {A,B} \right) = A' + B$$ Simplified expression for function $$f((x+y,y),z)$$ is

Transform the following logic circuit (without expressing its switching function) into an equivalent logic circuit that employs only $$6$$ $$NAND$$ ga...

Consider the following logic circuit whose inputs are functions $${f_1},$$ $${f_2},$$ $${f_3},$$ and output is $$f.$$ ...

The simultaneous equations on the Boolean variables $$x, y, z$$ and $$w,$$ $$$x+y+z=1$$$ $$$xy=0$$$ $$$xz+w=1$$$ $$$xy + \overline z \overline w = 0...

Which of the following sets of component(s) is/are sufficient to implement any arbitrary Boolean function?

Consider the logic circuit shown in Figure below. The functions $${f_1},$$ $${f_2}$$ and $$f$$ (in canonical sum of products form in decimal notation)...

Let $$f\left( {x,y,z} \right) = \overline x + \overline y x + xz$$ be a switching function. Which one of the following is valid?

Two $$NAND$$ gates having open collector outputs are tied together as shown in fig. The logic function $$Y,$$ implemented by the circuit is. ...

A circuit outputs a digit in the form of $$4$$ bits. $$0$$ is represented by $$0000$$, $$1$$ by $$0001..., $$ $$9$$ by $$1001.$$ A combinational circu...

Consider the following circuit composed of $$XOR$$ gates are non-inverting buffers. The non-inverting buffers have delays $${\delta _1} = 2$$ $$ns$$...

Express the function $$f( x, y, z)= xy'+ yz'$$ with only one complement operation and one or more $$AND/OR$$ operations. Draw the logic circuit implem...

A logic network has two data inputs $$A$$ and $$B,$$ and two control inputs $${C_0}$$ and $${C_1}$$. It implements the function $$F$$ according to the...

Find the minimum sum of products form of the logic function $$f\left( {A,B,C,D} \right) = \sum d \left( {3,11,12,14} \right)$$ Where $$m$$ and $$d$$...

Find the minimum product of sums of the following expression $$f = ABC + \overline A \,\overline B \,\overline C .$$

Show with the help of a block diagram represent Boolean function: $$f=AB+BC+CA$$ can be realized using only $$4:1$$ multiplexer.