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 $$š$$ and $$š$$ are Boolean variables. Consider the following two statemen...
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...
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 ...
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 = Ī£(0, 2, 5, 8, 14), f2 = Ī£(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... 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... 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...
Given the function $$F = Pā² + QR,$$ 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 $$f\left( {w,x,y,z} \right) = \sum {\left( {0,2,4,5,6,10} \right)}$$ _________________.
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
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