GATE CSE
Digital Logic
Boolean Algebra
Previous Years Questions

## Marks 1

Which one of the following is NOT a valid identity?
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 $$XOR... Consider the Boolean operator$$ \ne $$with the following properties:$$x \ne 0 = x,\,\,x \ne 1 = \overline x ,\,\,x \ne x = 0$$and$$x \ne \overli...
Let $$\ne$$ be a binary operator defined as $$X \ne Y = X' + Y'$$ where $$𝑋$$ and $$𝑌$$ are Boolean variables. Consider the following two statemen...
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 combinational function block involving four Boolean variables $$x, y, a,$$ $$b$$ where $$x, a, b$$ are inputs and $$y$$ is the...
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 ## Marks 2 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... 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 total number of prime implicants of the function $$f\left( {w,x,y,z} \right) = \sum {\left( {0,2,4,5,6,10} \right)}$$ _________________. 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...
The binary operator $$\ne$$ is defined by the following truth table. .tg {border-collapse:collapse;border-spacing:0;} .tg td{font-family:Arial, sa...
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...
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? $$F\left( {a,b,c} \right) = {a^1}c + a{c^1} + {b^1}c$$
Consider the following logic circuit whose inputs are functions $${f_1},$$ $${f_2},$$ $${f_3},$$ and output is $$f.$$ ...
Transform the following logic circuit (without expressing its switching function) into an equivalent logic circuit that employs only $$6$$ $$NAND$$ ga...
$$f\left( {A,B} \right) = A' + B$$ Simplified expression for function $$f((x+y,y),z)$$ is
The simultaneous equations on the Boolean variables $$x, y, z$$ and $$w,$$ $$x+y+z=1$$$$$xy=0$$$ $$xz+w=1$$$$$xy + \overline z \overline w = 0... Which of the following sets of component(s) is/are sufficient to implement any arbitrary Boolean function? Let$$f\left( {x,y,z} \right) = \overline x + \overline y x + xz$$be a switching function. Which one of the following is valid? Consider the logic circuit shown in Figure below. The functions$${f_1},{f_2}$$and$$f$$(in canonical sum of products form in decimal notation)... Two$$NAND$$gates having open collector outputs are tied together as shown in fig. The logic function$$Y,$$implemented by the circuit is. ... ## Marks 5 A circuit outputs a digit in the form of$$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} = 2ns$$... Express the function$$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$$... Show with the help of a block diagram represent Boolean function:$$f=AB+BC+CA$$can be realized using only$$4:1$$multiplexer. Find the minimum product of sums of the following expression$$f = ABC + \overline A \,\overline B \,\overline C .$\$
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