# GATE Exercises on Boolean Logic

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1 GATE Exerces on Boolean Logic 1 Abstract Th problem set has questions related to Boolean logic and gates taken from GATE papers over the last twenty years. Teachers can use the problem set for courses tutorials. 1) The Boolean expression(x + )(X + ) + (X)+X simplifies to a) X b) c) X d) X+ 2) The SOP (sum of products) form of a Boolean function (0, 1, 3, 7, ),where inputs are A,B,C,D(A MSB,and D LSB).The equivalent minimed expression of the function a) (B+ C)(A+ C)(A+B)(C+ D) b) (B+ C)(A+ C)(A+ C)(C+ D) c) (B+ C)(A+ C)(A+ C)(C+ D) d) (B+ C)(A+B)(A+ B)(C+D) 3) The Boolean expressionf(x,, Z) = XZ+ X.Z+ XZ+ XZ converted into the canonical product of sum (POS) form a) (X+ + Z)(X+ + Z)(X+ + Z)(X+ + Z)(X+ + Z) b) (X++Z)(X++Z)(X++Z)(X++Z) c) (X+ + Z)(X+ + Z)(X+ + Z) d) (X+ + )(X+ + Z)(X+ + Z) 4) If the functions W, X, and Z are as follows W = R + PQ + RS X = PQR.S + P.Q.R.S + PQ.R.S = RS + PR+ PQ+P.Q Z = R+S + PQ+P.Q.R+ PQ.S a) W= Z, X= Z b) W=Z, X= c) W= d) W== Z 5) The Boolean expression AC+BC equivalent to a) AC+BC b) BC+ AC+BC+ACB c) AC+BC+BC+ABC d) ABC+AB.C+ABC+A.B.C 6) The boolean expression = A.B.CD + ABCD+AB.CD+ ABC.D can be a) = A.B.CD+ ABC+ACD b) = A.B.CD+ BCD+ AB.CD c) = ABCD+ B.CD+ AB.CD d) = ABCD+ B.CD+ ABC.D 7) The Boolean expressionxz + X.Z + XZ+ XZ+XZ cab be simplified to a) XZ+ XZ+ Z b) XZ+ Z+ Z c) X+ Z+ XZ d) X+ Z+ XZ 8) The simplified form of the Boolean expression = (ABC+ D)(AD+ B.C) can be written as a) AD+B.CD b) AD+BCD c) (A+ D)(BC+ D) d) AD+BCD 9) A function of Boolean variables X, and Z expressed in terms of the min-terms as F(X,, Z)= (1, 2, 5, 6, 7) Which one of the product of sums given below equal to the function F(X,,Z)? a) (X+ + Z)(X+ + Z)(X+ + Z)

2 b) (X+ + Z)(X+ + Z)(X+ + Z) c) (X+ + Z)(X+ + Z(X+ + Z)(X+ + Z)(X+ + Z) d) (X+ + Z)(X+ + Z)(X+ + Z)(X+ + Z)(X+ + Z) ) For the logic circuit shown in Fig., the required input condition (A,B,C) to make the output (X)=1 A B C C Fig. a) 1, 0, 1 b) 0, 0, 1 c) 1, 1, 1 d) 0, 1, 1 ) For the logic circuit shown in Fig., the simplified Boolean expression for the output A B C Fig. a) A+B+C b) 0 c) 1 d) C 12) If X=1 in logic equation [X+ Z+ (Z+X)]X+ Z(X+ ) = 1, then a) =Z b) = Z c) Z= 1 d) Z= 0 13) The output in the circuit below always 1 when P Q R P Fig. 13 a) two or more of the inputs P,Q,R are 0 b) two or more of the inputs P,Q,R are 1 c) any odd number of the inputs P,Q,R 0 d) any odd number of the inputs P,Q,R 1 14) In the sum of product function f(x,,z)= (2, 3, 4, 5),prime implicants are a) X, X b) X, X.Z, XZ c) XZ, XZ, X d) XZ, XZ, X.Z, XZ 15) The logic function f=(x.) + (X.) same as a) (X+ )(X+ ) b) (X+ )+(X+ ) c) (X).(X.) d) None of the above 16) The minimal product of sums function described by the K-map given in Fig. a) A.C b) A+C c) A+C d) AC 17) For the circuit shown in Fig., the Boolean expression for the output in terms of inputs P, Q,R and S 2

3 P Q R S AB C x x 0 Fig. 16 Fig. 17 a) P+ Q+R+S b) P+ Q+R+S c) (P+Q)(R+S ) d) (P+Q)(R+S ) 18) The following Karnaugh map represents a function F(X,,Z),minimized form of the function F Z X b) X.+ Z c) X+ Z d) X+ Z 19) The output of the logic circuit given below X Fig. 19 a) 1 b) 0 c) X d) X 20) The Boolean expression AB+AC+BC simplifies to a) BC+AC b) AB+AC+B c) AB+AC d) AB+ BC 21) The output expression for the Karnaugh map shown below CD AB Fig. 21 a) X+ Z Fig. 18 a) BD+ BCD b) BD+AB c) BD+ABC d) BD+ABC 3

4 22) For 3 input logic circuit shown below,the output Z can be expressed as P Q R Fig. 22 a) Q+R b) PQ+R c) Q+R d) P+ Q+R 23) Given f (W, X,, Z) = m(0, 1, 2, 3, 7, 8, )+ d(5, 6,, 15), where d represent don t-care condition in Karnaugh maps.which of the following a minimum product-of-sum(pos)form of f(x,,z,w)? a) (W+ Z)(X+ Z) b) (W+ Z)(X+ Z) c) (W+ Z)(X+ Z) d) (W+ Z)(X+ Z) 24) The output expression for the Karnaugh map shown below BC A a) A+ B b) A+C c) A+C d) A+C Fig. 24 Z 25) The Boolean expression (a+b+c+d)+(b+c) simplifies to a) 1 b) ab c) ab d) 0 26) f (A, B, C, D) = ΠM(0, 1, 3, 4, 5, 7, 9,, 12, 13, 14, 15) a maxterm representation of a Boolean function f(a,b,c,d) where A the MSB and D the LSB.The equivalent minimized representation of th function a) (A+C+ D)(A+ B+ D) b) ACD+ ABD c) ACD+ABCD d) (B+C+D)(A+B+C+D)(A+B+C+D) 27) Consider the following Sum of Products expression, F. F= ABC+ A.BC+ ABC+ ABC+A.B.C The equivalent Product of Sums expression a) (A+ B+C)(A+ B+C)(A+ B+C) b) (A+ B+C)(A+ B+C)(A+ B+C) c) (A+ B+C)(A+ B+C)(A+ B+C) d) (A+ B+C)(A+ B+C)(A+ B+C) 28) The number of product term in the minimized sum-of-product expression obtained through the following K-map (where d denote don t-care states) a) 2 b) 3 c) 4 d) 5 29) The minimum sum of products form of the Boolean expression = P.Q.R.S+ PQ.R.S+ PQ.RS+ PQRS + a) PQ+Q.S PQRS + P.Q.RS 4

5 d d equivalent to a) XZ+ X.Z b) X..Z + XZ c) (X+ Z)(+ Z) d) (X+ Z)(+ Z) 33) In the digital circuit given below, F X F Fig. 28 b) PQ+QRS c) PQ+Q.R.S d) Q.S+ PQR 30) A logic circuit implements the Boolean function F= X+ X.Z. It found that the input combination X = = 1 can never occur. Taking th into account, a simplified expression for F, a) X+.Z b) X+ Z c) X+ d) + XZ 31) The logic evaluated by the circuit at the output X Fig. 31 Output a) X+ X b) (X+ )X c) (X)+ X d) X+ X+ X+ 32) The Boolean expression X+ (X+ )Z Z Fig. 33 a) X+ Z b) X+ Z c) X. d) XZ+ 34) For the Boolean expression f = a.b.c + abc+ab.c+abc+abc,the minimized Product of Sum (PoS) expression a) f= (b+c)(a+c) b) f= (b+c)(a+c) c) f= (b+c)(a+c) d) f= c+abc 35) The minimal sum-of-products expression for the logic function f represented by the given Karnaugh map a) QS+ PRS+ PQR+PRS+ PQR b) QS+ PRS+ P.QR+PR.S+ PQR c) PRS + P.Q.R d) PRS+ PQR+PRS+ PQR 36) The expression A+AB equivalent to a) A+ B b) A+ B c) AB+A d) AB 37) The simplest form of the Boolean expression ABC.D+ABCD+ABCD+ABCD 5

6 PQ RS Fig. 35 a) AD b) BC c) AB d) AB 38) The Karnaugh map for a four variable Boolean function given in Fig. PQ RS Boolean variables x, y, z and w, x+y+z=1 xy=0 xz+w=1 xy+z.w=0 have following solution for x,y,z and w,respectively: a) 00 b) c) d) 00 40) If P, Q, R are Boolean variables, then (P+ Q)(PQ+PR)(P.R+ Q) Simplifies to a) PQ b) PR c) PQ+R d) PR+ Q 41) Which functions does NOT implement the Karnaugh map given below? a) (w+ x)y b) xy+yw c) (w+ x)(w+y)(x+y) d) Noneo f theabove WZ X 00 0 x 0 0 Fig. 38 a) PQRS+ QS b) PQRS+ QS c) PQR+QS d) PQRS+ Q 39) The simultaneous equations on the 6 0 x x 0 0 Fig. 41

7 42) Given the following Karnaugh map, which one of the following represents the minimal Sum-Of-Products of the map? WX Z 00 0 x 0 x x 1 x 1 0 x x 0 Fig. 42 a) X+ Z b) WX.+ X+ XZ c) WX+ Z+X d) XZ+ 43) Minimum sum of product expression for f(w,x,y,z) shown in Karnaugh-map below wx yz a) XZ+ Z x x x 1 Fig. 43 b) XZ+ ZX c) X+ ZX d) None of the above 44) The minterm expansion of f (P, Q, R) = PQ+QR+ PR a) m 2 + m 4 + m 6 + m 7 b) m 0 + m 1 + m 3 + m 5 c) m 0 + m 1 + m 6 + m 7 d) m 2 + m 3 + m 4 + m 5 45) In the Karnaugh map shown below, X denotes a don t care term. What the minimal form of the function represented by the Karnaugh map? ab cd x x 1 1 x Fig. 45 a) b.d+a.d b) a.b+b.d+abd c) b.d+abd d) a.b+b.d+a.d 46) The simplified SOP (Sum of Product) form of the Boolean expression (P+Q+R)(P+Q+R)(P+ Q+R) a) PQ+R b) P+ Q.R c) PQ+R d) PQ+R 47) Which one of the following gives the 7

8 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 and m 5 are minterms corresponding to the input A, B and C, with A as MSB and C as the LSB. a) ĀB+Ā B C+ A BC b) Ā C+ĀB+ A BC c) Ā C+A B+ A BC d) ĀBC+Ā C+A BC 48) What the minimal form of the Karnaugh map shown below? Assume that X denotes a don t care term. a) b.d b) b.d+b.c c) b.d+ab.cd d) b.d+b.c+c.d 50) The truth table represents the Boolean function X f(x,) a) X b) X+ c) X d) ab cd 00 1 x x 1 x 1 1 x Fig ) Consider the Boolean function,f(w, X,, Z) = W + X + WXZ+ W X+ XZ+ XȲ Z. Which of the following the complete set of essential prime implicants? a) W,, XZ, X Z b) W,, XZ c), XȲ Z d), XZ, X Z 8

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