1(a) Ans.
State and verify De Morgan’s law in Boolean Algebra.
DeMorgan’s theorems state that (i) (X + Y)’= X’.Y’ (ii) (X.Y)’= X’ + Y’
(i) (X + Y)’= X’.Y’
Now to prove DeMorgan’s first theorem, we will use complementarity laws.
Let us assume that P = x + Y where, P, X, Y are logical variables. Then, according to complementation law P + P’ =1 and P . P’= 0
That means, if P, X, Y are Boolean variables hen this complementarity law must hold for variables P. In other words, if P i.e., if (X + Y)’= X’.Y’then
(X + Y) + (XY)’must be equal to 1. (as X + X’= 1) (X + Y) . (XY)’must be equal to 0. (as X . X’= 0) Let us prove the first part, i.e.,
(X + Y) + (XY)’ = 1
(X + Y) + (XY)’= ((X + Y) +X’).((X + Y) +Y’) (ref. X + YZ = (X + Y)(X + Z))
= (X + X’+ Y).(X + Y +Y’)
= (1 + Y).(X + 1) (ref. X + X’=1)
=1.1 (ref. 1 +X =1)
= 1 So first part is proved.
Now let us prove the second part i.e., (X + Y) . (XY)’= 0
(X + Y) . (XY)’ = (XY)’ . (X + Y) (ref. X(YZ) = (XY)Z)
= (XY)’X + (XY)’Y (ref. X(Y + Z) = XY + XZ)
= X(XY)’ + X’YY’
= 0 .Y + X’ . 0 (ref. X . X’=0)
= 0 + 0 = 0
So, second part is also proved, Thus: X + Y = X’ . Y’
(ii) (X.Y)’= X’ + Y’
Again to prove this theorem, we will make use of complementary law i.e., X + X’= 1 and X . X’= 0
If XY’s complement is X + Y then it must be true that (a) XY + (X’+ Y’) = 1 and (b) XY(X’+ Y’) = 0 To prove the first part
L.H.S = XY + (X’+Y’)
= (X’+Y’) + XY (ref. X + Y = Y + X)
= (X’+Y’ + X).(X’+Y’ + Y) (ref. (X + Y)(X + Z) = X + YZ)
= (X + X’+Y’).(X’ + Y +Y’)
= (1 +Y’).(X’ + 1) (ref. X + X’=1)
=1.1 (ref.1+X=1)
= 1 = R.H.S Now the second part i.e.,
XY.(X+ Y) = 0
L.H.S = (XY)’.(X’+Y’)
= XYX’ + XYY’ (ref. X(Y + Z) = XY + XZ)
= XX’Y + XYY’
= 0.Y + X.0 (ref. X . X’=0)
= 0 + 0 = 0 = R.H.S.
XY.(X’ + Y’)= 0 and XY + (Xʹ +Y’) = 1 (XY)’= X’ + Y’. Hence proved.
1(b).
Ans.
Draw a Logical Circuit Diagram for the following Boolean Expression X’.(Y’ + Z)
The Logical Circuit Diagram for the Boolean Expression is as following:
1(c) Ans.
Convert the following Boolean expression into its equivalent Canonical Sum of Product F orm (SOP).
(X’ + Y + Z’).(X’ + Y + Z).(X’ + Y’ + Z’) Given (X’ + Y + Z’).(X’ + Y + Z).(X’ + Y’ + Z’)
(1 + 0 + 1) (1 + 0 + 0) (1 + 1 + 1)
= M4.M5.M6.M7
= π(4, 5, 6, 7)
SOP is equal to(excluding position of minterms)
= ∑(0, 1, 2, 3)
= m0+ m1+ m2+ m3
= X’Y’Z + X’Y’Z + X’YZ’ + X’YZ 1(d)
Ans.
Reduce the following Boolean expression using K-map F(A, B, ,C, D) = ∑(0,2,3,4,5,6,7,8,10,12)
There are 1 Pair and 2 Quads that reduce as given below:
Pair(m8+ m10) reduces to AB’D’
Quad-1(m0+ m4+ m12+ m8) reduces to C’D’
Quad- 2(m2+ m3+ m6+ m7) reduces to A’C Simplified Boolean expression for given K-map is
F(A,B,C,D) = AB’D’ + C’D’ + A’C 2(a).
Ans.
State De Morgan’s Theorems and verify the same using truth table.
De Morgan’s First theorem. It states that (X+Y)’=X’.Y’
Truth Table for first theorem.
X Y X’ Y’ X+Y (X+Y)’ X’.Y’
0 0 1 1 0 1 1
0 1 1 0 1 0 0
De Morgan’s Second theorem. It states that (X.Y)’=X’+Y’
Truth Table for second theorem.
X Y X’ Y’ X.Y (X.Y)’ X’+Y’
0 0 1 1 0 1 1
0 1 1 0 0 1 1
1 0 0 1 1 0 0
1 1 0 0 1 0 0
From Truth Table it is proved that (X+Y)’ = X’.Y’
1 0 0 1 0 1 1
1 1 0 0 1 0 0
From Truth Table it is proved that (X.Y)’ = X’ + Y’
2(b).
Ans.
Write the equivalent Canonical Product of Sum Expression for the following Sum of Product Expression : F(X, Y, Z)
= ∑(0, 2, 4, 5)
Given SOP expression : F(X, Y, Z) = ∑(0, 2, 4, 5) Equivalent POS expression : π(1, 3, 6, 7)
Equivalent POS expression will be = M1.M3.M6.M7
M1 = (0 + 0 + 1)’S Maxterm = (X + Y + Z) M3 = (0 + 1 + 1)’S Maxterm = (X + Y’ + Z’) M6 = (1 + 1 + 0)’S Maxterm = (X’ + Y’ + Z) M7 = (1 + 1 + 1)’S Maxterm = (X’ + Y’ + Z’)
∵Equivalent POS will be
F(X, Y, Z) = (X + Y + Z). (X + Y’ + Z’). (X’ + Y’ + Z). (X’ + Y’ + Z’) 2(c).
Ans.
Write the equivalent Boolean expression for the following Logic Circuit.
The equivalent Boolean expression for the given Logic C ircuit is: F = AB’ + C’D 2(d).
Ans.
Reduce the following Boolean expression using K-map : F(A, B, C, D) = ∏(5, 6, 7, 8, 9, 12, 13, 14, 15)
There are 1 Pair and 2 Quads that reduce as given below:
Pair(M5 . M7) reduces to (A + B’ + D’)
Quad-1(M6. M7. M14. M15) reduces to (B’ + C’) Quad- 2(M8. M9. M12. M13) reduces to (A’ + C) Simplified Boolean expression for given K-map is
F(A,B,C,D) = (A + B’ + D’). (B’ + C’). (A’ + C) 3(a).
Ans.
State DeMorgan’s Laws. Verify them using truth t ables.
De Morgan’s First theorem. It states that (X+Y)’=X’.Y’
Truth Table for first theorem.
X Y X’ Y’ X+Y (X+Y)’ X’.Y’
0 0 1 1 0 1 1
0 1 1 0 1 0 0
1 0 0 1 1 0 0
1 1 0 0 1 0 0
From Truth Table it is proved that (X+Y)’ = X’.Y’
De Morgan’s Second theorem. It states that (X.Y)’=X’+Y’
Truth Table for second theorem.
X Y X’ Y’ X.Y (X.Y)’ X’+Y’
0 0 1 1 0 1 1
0 1 1 0 0 1 1
1 0 0 1 0 1 1
1 1 0 0 1 0 0
From Truth Table it is proved that (X.Y)’ = X’ + Y’
3(b).
Ans.
Prove (A + B).(A’ + C)=(A + B + C).(A + B + C’).(A’ + B + C).(A’ + B’ + C) algebraically.
RHS = (A + B + C).(A + B + C’).(A’ + B + C).(A’ + B’ + C)
= (A + B)(C + C’).(A’ + C)(B + B’) (∵C + C’=1, B + B’=1)
= (A + B).(A’ + C)
= LHS 3(c).
Ans.
Obtain a simplified form for a Boolean expression :
F(X, Y, Z, W) = ∑(0, 1, 4, 5, 7, 8, 9, 12, 13, 15) using K-map method.
There are 1 Pair and 1 Octet that reduce as given below:
Pair(m7+ m15) reduces to YZW
Octet(m0+ m1+ m4+ m5+ m8+ m9+ m12+ m13) reduces to Z’
Simplified Boolean expression for given K-map is F(X,Y,Z,W) = YZW + Z’
3(d). Draw the circuit diagram for the Boolean function F(X, Y, Z) = (X’ + Y)(Y’ + Z) using NOR gates only.
Ans. The circuit diagram for the given Boolean function is as following:
3(e).
Ans.
Express in the POS form, the Boolean function F(A, B, C), the truth table for which is given below :
A B C F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 0
1 1 1 1
The desired Canonical Product-of-Sum form is as following;
F = π(0, 2, 4, 6) = (A + B + C)(A + B’ + C)(A’ + B + C)(A’ + B’ + C) 4(a).
Ans.
State the distributive law. Verify the law using truth table.
Distributive law state that (a) X(Y +Z) = XY + XZ (b) X + YZ = (X + Y)(X + Z) (a) X(Y +Z) = XY + XZ
To prove this law, we will make a following truth table :
X Y Z Y+Z XY XZ X(Y+Z) XY+XZ
0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0
0 1 0 1 0 0 0 0
0 1 1 1 0 0 0 0
1 0 0 0 0 0 0 0
1 0 1 1 0 1 1 1
1 1 0 1 1 0 1 1
1 1 1 1 1 1 1 1
From truth table it is prove that X(Y +Z) = XY + XZ (b) X + YZ = (X + Y)(X + Z)
X Y Z YZ X + YZ XZ X + Y X + Z (X + Y)(X + Z)
0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 1 0
0 1 0 0 0 0 1 0 0
0 1 1 1 1 0 1 1 1
1 0 0 0 1 0 1 1 1
1 0 1 0 1 1 1 1 1
1 1 0 0 1 0 1 1 1
1 1 1 1 1 1 1 1 1
From truth table it is prove that X + YZ = (X + Y)(X + Z) 4(b).
Ans.
Prove x + x’y = x + y algebraically.
LHS = x + x’y
= (x + x’)(x + y) (X + YZ = (X + Y)(X + Z) Distributive law)
=x+y (∵x + x’ =1)
= RHS 4(c).
Ans.
Write the dual of the Boolean expression (x + y).(x’ + y’) The dual of the given Boolean ex pression is xy + x’y’
4(d).
Ans.
Minimize F(w, x, y, z) using Karnaugh map F(w, x, y, z) = ∑(0, 4, 8, 12)
There is 1 Quad(m0+ m4+ m8+ m12) that reduces to y’z’
Simplified Boolean expression for given K-map is F(w, x, y, z) = y’z’
4(e).
Ans.
Represent the Boolean expression (x + y)(y + z)(z + x) with the help of NOR gates only.
4(f).
Ans.
Write sum of products form of function F(x, y, z). The truth table representation for the function F is given below :
x y z F
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 0
1 1 1 1
The desired Canonical Sum-of-Product form is as following;
F = ∑(2, 4, 7) = x’yz’ + xy’z’ + xyz 5(a).
Ans:
State and prove DeMorgan’s Theorem (Any One) algebracaly.
DeMorgan’s theorems state that (i) (X + Y)’= X’.Y’ (ii) (X.Y)’= X’ + Y’
(i) (X + Y)’= X’.Y’
Now to prove DeMorgan’s first theorem, we will use complementarity laws.
Let us assume that P = x + Y where, P, X, Y are logical variables. Then, according to complementation law P + P’ =1 and P . P’= 0
That means, if P, X, Y are Boolean variables hen this complementarity law must hold for variables P. In other words, if P i.e., if (X + Y)’= X’.Y’then
(X + Y) + (XY)’must be equal to 1. (as X + X’= 1) (X + Y) . (XY)’must be equal to 0. (as X . X’= 0) Let us prove the first part, i.e.,
(X + Y) + (XY)’ = 1
(X + Y) + (XY)’= ((X + Y) +X’).((X + Y) +Y’) (ref. X + YZ = (X + Y)(X + Z))
= (X + X’+ Y).(X + Y +Y’)
= (1 + Y).(X + 1) (ref. X + X’=1)
=1.1 (ref. 1 +X =1)
= 1 So first part is proved.
Now let us prove the second part i.e., (X + Y) . (XY)’= 0
(X + Y) . (XY)’ = (XY)’ . (X + Y) (ref. X(YZ) = (XY)Z)
= (XY)’X + (XY)’Y (ref. X(Y + Z) = XY + XZ)
= X(XY)’ + X’YY’
= 0 .Y + X’ . 0 (ref. X . X’=0)
= 0 + 0 = 0
So, second part is also proved, Thus: X + Y = X’ . Y’
5(b). Given the following truth table, driven a Sum of Product (SOP) and Product of Sum (POS) form of Boolean
Ans.
expression from it :
X Y Z G(X,Y,Z)
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 0
1 1 1 1
The desired Canonical Sum-of-Product form is as following;
G = ∑(1, 2, 5, 7) = X’Y’Z + X’YZ’ + XY’Z + XYZ The desired Canonical Product-of-Sum form is as following;
G = π(0, 3, 4, 6) = (X + Y + Z)(X + Y’ + Z’)(X’ + Y + Z)(X’ + Y’ + Z) 5(c).
Ans.
Obtain a simplified form for the following Boolean Expression using Karnaugh’s Map : F(u, v, w, z) = ∑(0, 3, 4, 5, 7, 11, 13, 15).
There are 2 Pairs and 1 Quad that reduce as given below:
Pair-1(m0+ m4) reduces to u’w’z’
Pair-2(m5+ m13) reduces to vw’z
Quad(m3+ m7+ m11+ m15) reduces to wz
Simplified Boolean expression for given K-map is F(u, v, w, z) = u’w’z’ + vw’z + wz
5(d).
Ans.
Draw the logic circuit for a Half Adder using NOR gates only.
5(e).
Ans.
Interpret the following Logic Circuit as Boolean Expression :
The equivalent Boolean expression for the given Logic Circuit is: F = (W + X’)(Y’ + Z) 6(a).
Ans.
State DeMorgan’s Laws. Verify one of the DeMorgan’s laws using truth tables.
De Morgan’s First theorem. It states that (X+Y)’=X’.Y’
De Morgan’s Second theorem. It states that (X.Y)’=X’+Y’
Truth Table for first theorem.
X Y X’ Y’ X+Y (X+Y)’ X’.Y’
0 0 1 1 0 1 1
0 1 1 0 1 0 0
1 0 0 1 1 0 0
1 1 0 0 1 0 0
From Truth Table it is proved that (X+Y)’ = X’.Y’
6(b).
Ans.
Prove X + Y’Z = (X + Y’ + Z’)(X + Y’ + Z)(X + Y + Z) algebraically.
Try by Yourself.
6(c).
Ans.
Write the dual of the Boolean expression (U + W)(V’U + W) The dual of the given Boolean ex pression is UW + (V’ + U)W 6(d). Obtain a simplified form for a Boolean expression :
F(u, v, w, z) = ∑(0, 1, 3, 5, 7, 9, 10, 11, 12, 13, 14, 15( using Karnaugh Map.
Ans. There are 3 Pairs and 1 Octet that reduce as given below:
Pair-1(m0+ m1) reduces to u’v’w’
Pair-2(m12+ m13) reduces to uvw’
Pair-3(m10+ m14) reduces to uwz’
Octet(m1+ m3+ m5+ m7+ m9+ m11+ m13+ m15) reduces to z Simplified Boolean expression for given K-map is
F(u, v, w, z) = u’v’w’ + uvw’ + uwz’ + z 6(e).
Ans.
Represent the Boolean expression X + Y . Z’ with the help of NOR gates only.
6(f).
Ans.
Write the Product of Sum form of the function H(U, V, W), truth table representation of H is as follows :
U V W H
0 0 0 1
0 0 1 0
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 0
1 1 1 1
The desired Canonical Product-of-Sum form is as following;
H = π(1, 3, 4, 6) = (U + V + W’)(U + V’ + W’)(U’ + V + W’)(U’ + V’ + W) 7(a).
Ans.
State and prove the absorption law algebraically.
Absorption law states that (i) X + XY = X and (ii) X(X + Y) = X (i) X + XY = X
LHS = X + XY = X(1 + Y)
= X . 1 [∵1 + Y = 1]
= X = RHS. Hence proved.
(ii) X(X + Y) = X
LHS = X(X + Y) = X . X + XY
= X + XY
= X(1 + Y)
= X . 1
= X = RHS. Hence proved.
7(b).
Ans.
Given the following truth table, derive a sum of product (SOP) and Product of Sum (POS) form of Boolean expression from it :
A B C G(A,B,C)
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 0
1 1 1 1
The desired Canonical Sum-of-Product form is as following;
G = ∑(1, 2, 5, 7) = A’B’C + A’BC’ + AB’C + ABC The desired Canonical Product-of-Sum form is as following;
G = π(0, 3, 4, 6) = (A + B + C)(A + B’ + C’)(A’ + B + C)(A’ + B’ + C) 7(c).
Ans.
Obtain a simplified form for the following Boolean Expression using Karnaugh’s Map : F(a, b, c, d) = ∑(0, 1, 2, 4, 5, 7, 8, 9, 10, 11, 14)
There are 3 Pairs and 2 Quads that reduce as given below:
Pair-1(m0+ m2) reduces to a’b’d’
Pair-2(m5+ m7) reduces to a’b’d Pair-3(m12+ m14) reduces to acd’
Quad-1(m0+ m1+ m4+ m5) reduces to a’c’
Quad-2(m8+ m9+ m10+ m11) reduces to ab’
Simplified Boolean expression for given K-map is F(a, b, c, d) = a’b’d’ + a’b’d + acd’ + a’c’ + ab’
7(d).
Ans.
Draw the logic circuit for a Half Adder using NAND gates only.
7(e).
Ans.
Interpret the following Logic Circuit as Boolean Expression :
The equivalent Boolean expression for the given Logic Circuit is: F = (W + X’).(Y’ + Z) 8(a).
Ans.
State Absorption Laws. Verify one of the Absorption Law using truth table.
Absorption law states that (i) X + XY = X and (ii) X(X + Y) = X Truth Table for X + XY = X
X Y XY X+XY
0 0 0 0
0 1 0 0
1 0 0 1
1 1 1 1
From Truth Table it is proved that X + XY = X 8(b).
Ans.
Verify X.Y’ + Y’.Z = X.Y’.Z + X.Y’.Z’ + X’.Y’.Z algebraically.
RHS = X.Y’.Z + X.Y’.Z’ + X’.Y’.Z
= X.Y’(Z + Z’) + X’.Y’.Z
= X.Y’ + X’.Y’.Z
= X.Y’ + Y’(X + X’.Z) ( X + X’ = 1)
= X.Y’ + Y’.Z
= LHS 8(c).
Ans.
Write the dual of the Boolean expression A + B’ . C The dual of the given Boolean ex pression is A.(B’ + C) 8(d).
Ans.
Obtain a simplified form for a boolean expression
F(U, V, W, Z) = ∑(0, 1, 3, 4, 5, 6, 7, 9, 10, 11, 13, 15) using Karnaugh Map.
There are 3 Pairs and 1 Octet that reduce as given below:
Pair-1(m0+ m4) reduces to U’W’Z’
Pair-2(m6+ m7) reduces to U’VW Pair-3(m10+ m11) reduces to UV’W
Octet (m1+ m3+ m5 + m7 +m9+ m11+ m13+ m15) reduces to Z Simplified Boolean expression for given K-map is
F(U, V, W, Z) = U’W’Z’ + U’VW + UV’W + Z 8(e). Represent the Boolean expression X . Y’ + Z with the help of NOR gates only.
Ans.
8(f).
Ans.
Write the Product of Sum form of the function H(U, V, W), truth table representation of H is as follows :
U V W H
0 0 0 1
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 0
1 1 1 0
The desired Canonical Product-of-Sum form is as following;
H = π(2, 4, 6, 7) = (U + V’ + W)(U’ + V + W)(U’ + V’ + W)(U’ + V’ + W’) 9(a).
Ans.
State the distributive law and verify the law using truth table.
Distributive law state that (a) X(Y +Z) = XY + XZ (b) X + YZ = (X + Y)(X + Z) (a) X(Y +Z) = XY + XZ
To prove this law, we will make a following truth table :
X Y Z Y+Z XY XZ X(Y+Z) XY+XZ
0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0
0 1 0 1 0 0 0 0
0 1 1 1 0 0 0 0
1 0 0 0 0 0 0 0
1 0 1 1 0 1 1 1
1 1 0 1 1 0 1 1
1 1 1 1 1 1 1 1
From truth table it is prove that X(Y +Z) = XY + XZ (b) X + YZ = (X + Y)(X + Z)
X Y Z YZ X + YZ XZ X + Y X + Z (X + Y)(X + Z)
0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 1 0
0 1 0 0 0 0 1 0 0
0 1 1 1 1 0 1 1 1
1 0 0 0 1 0 1 1 1
1 0 1 0 1 1 1 1 1
1 1 0 0 1 0 1 1 1
1 1 1 1 1 1 1 1 1
From truth table it is prove that X + YZ = (X + Y)(X + Z) 9(b).
Ans.
Prove XY + YZ + Y’Z = XY + Z, algebraically.
LHS = XY + YZ + Y’Z
= XY + Z(Y + Y’) ( Y + Y’ = 1)
= XY + Z 9(c).
Ans.
Obtain the simplified form of a boolean expression using Karnaugh map “ F(w, x, y, z) = ∑(2, 3, 6, 10, 11, 14)
There are 1 Pair and 1 Quad that reduce as given below:
Pair (m3+ m11) reduces to x’yz
Quad (m2+ m6+ m10+ m14) reduces to yz’
Simplified Boolean expression for given K-map is F(w, x, y, z) = x’yz + yz’
9(d).
Ans.
Represent the Boolean expression (X + Y)(Y + Z)(X + Z) with help of NOR gate only.
9(e).
Ans.
Given the following truth table, write the products of sums form of the function F(x, y, z):
x y z F
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 0
1 1 1 1
The desired Canonical Product-of-Sum form is as following;
F = π(0, 3, 4, 6) = (x + y + z)(x + y’ + z’)(x’ + y + z’)(x’ + y’ + z) 10(a). State and verify Duality principle.
The principle of duality states that starting with a Boolean relation, another Boolean relation can be derived by : 1 .Changing each OR sign(+) to an AND sign(.).
2. Changing each AND sign(.) to an OR sign(+).
3. Replacing each 0 by 1 and each 1 by 0.
For example postulate II states (a) 0 + 0 = 0 (b) 0 + 1 = 1 (c) 1 + 0 = 1 (d) 1 + 1 = 1 Now according to principle of duality we changed ‘+’ to ‘.’ and 0 to 1.
These become (i) 1 . 1 = 1 (ii) 1 . 0 = 0 (iii) 0 . 1 = 0 (iv) 0 . 0 = 0 ,which are same as postulate III.
So i, ii, iii, iv are duals of a, b, c, d.
10(b).
Ans.
Prove algebraically x’y’z’ + x’y’z + x’yz + x’yz’ + xy’z’ + xy’z = x’ + y’
LHS = x’y’z’ + x’y’z + x’yz + x’yz’ + xy’z’ + xy’z
= x’y’(z’ + z) + x’y(z + z’) + xy’(z’ + z) ( z’ + z = 1, z + z’ = 1)
= x’y’ + x’y + xy’
= x’(y’ + y) + xy’
= x’ + xy’ ( y’ + y = 1)
= x’ + (x’)’y’ ( x = (x’)’)
= x’ + y’ ( a + a’b = a + b x’ + xy’ = x’ + y’)
= RHS 10(c).
Ans.
If F(a, b, c, d) = ∑(0, 1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15), obtain the simplified form using K-Map.
There are 1 Quad and 1 Octet that reduce as given below:
Pair (m3+ m7+ m11+ m15) reduces to cd
Quad (m0+ m1+ m4+ m5+ m8+ m9+ m12+ m13) reduces to c’
Simplified Boolean expression for given K-map is F(a, b, c, d) = cd + c’
10(d). Seven inverters are cascaded one after another. What is the output if the input is 1?
Ans. 0 10(e).
Ans.
Given the following circuit :
What is the output if (i) both inputs are FALSE
(ii) one is FALSE and the other is TRUE ? (i) FALSE (ii) FALSE
11(a).
Ans.
State and verify Absorption law in Boolean Algebra.
Absorption law states that (i) X + XY = X and (ii) X(X + Y) = X Truth Table for X + XY = X
X Y XY X+XY
0 0 0 0
0 1 0 0
1 0 0 1
1 1 1 1
From Truth Table it is proved that X + XY = X
Truth Table for X(X + Y) = X
X Y X +Y X(X + Y)
0 0 0 0
0 1 1 0
1 0 1 1
1 1 1 1
From Truth Table it is proved that X(X + Y) = X 11(b).
Ans.
Draw a Logical Circuit Diagram for the following Boolean Expression : A . (B + C’)
11(c).
Ans.
Convert the following Boolean expression into its equivalent Canonical Product of Sum Form(POS) : A.B’.C + A’.B.C + A’.B.C’
Given A.B’.C + A’.B.C + A’.B.C’
(1 0 1) (0 1 1) (0 1 0)
= m5+ m3+ m2
= ∑(2, 3, 5)
POS is equal to (excluding positions of minterms)
= π(0, 1, 4, 6, 7)
= M0.M1.M4.M6.M7
= (A + B + C).(A + B + C’).(A’ + B + C).(A’ + B’ +C).(A’ + B’ + C’) 11(d).
Ans.
Reduce the following Boolean expression using K-map:
F(A, B, C, D) = ∑(0, 1, 2, 4, 5, 8, 9, 10, 11)
There are 1 Pair and 2 Quad that reduce as given below:
Pair (m2+ m10) reduces to B’CD’
Quad-1 (m0+ m1+ m4+ m5) reduces to A’C’
Quad-2 (m0+ m1+ m4+ m5) reduces to AB’
Simplified Boolean expression for given K-map is F(A, B, C, D) = B’CD’ +’ A’C’ + AB’
12(a).
Ans.
State and verify Distributive law in Boolean Algebra.
Distributive law state that (a) X(Y +Z) = XY + XZ (b) X + YZ = (X + Y)(X + Z) (a) X(Y +Z) = XY + XZ
To prove this law, we will make a following truth table :
X Y Z Y+Z XY XZ X(Y+Z) XY+XZ
0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0
0 1 0 1 0 0 0 0
0 1 1 1 0 0 0 0
1 0 0 0 0 0 0 0
1 0 1 1 0 1 1 1
1 1 0 1 1 0 1 1
1 1 1 1 1 1 1 1 From truth table it is prove that X(Y +Z) = XY + XZ
(b) X + YZ = (X + Y)(X + Z)
X Y Z YZ X + YZ XZ X + Y X + Z (X + Y)(X + Z)
0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 1 0
0 1 0 0 0 0 1 0 0
0 1 1 1 1 0 1 1 1
1 0 0 0 1 0 1 1 1
1 0 1 0 1 1 1 1 1
1 1 0 0 1 0 1 1 1
1 1 1 1 1 1 1 1 1
From truth table it is prove that X + YZ = (X + Y)(X + Z) 12(b).
Ans.
Draw a Logical Circuit Diagram for the following Boolean Expression: A’.(B + C)
12(C).
Ans.
Convert the following Boolean expression into its equivalent Canonical Sum of Product Form(SOP).
(U’ + V’ + W’).(U + V’ + W’).(U + V + W) Given (U’ + V’ + W’).(U + V’ + W’).(U + V + W)
(1 + 1 + 1) (0 + 1 + 1) (0 + 0 + 0)
= M0.M3.M7
= π(0, 3, 7)
SOP is equal to(excluding position of Maxterms)
= ∑(1, 2, 4, 5, 6)
= m1+ m2+ m4+ m5+ m6
= U’V’W + U’VW’ + UV’W’ + UV’W + UVW’
12(d).
Ans.
Reduce the following Boolean expression using K-map:
F(A, B, C, D) = ∑(0, 3, 4, 5, 7, 9, 11, 12, 13, 14)
There are 4 Pair and 1 Quad that reduce as given below:
Pair-1(m4+ m5) reduces to A’BC’
Pair-2(m7+ m13) reduces to A’CD Pair-3(m9+ m11) reduces to AB’D Pair-4(m12+ m14) reduces to ABD’
Quad(m1+ m5+ m9+ m13) reduces to C’D
Simplified Boolean expression for given K-map is F(A, B, C, D) = A’BC’ +’ A’CD + AB’D + ABD’ + C’D 13(a).
Ans.
Verify the following algebraically: X’.Y + X.Y’ = (X’ + Y’).(X + Y) RHS = (X’ + Y’).(X + Y)
= (X’ + Y’).X + (X’ + Y’).Y
= X’.X + X.Y’ + X’.Y + Y’.Y
= 0 + X.Y’ + X’.Y + 0
= X.Y’ + X’.Y
= LHS (Verified) 13(b).
Ans.
Write the equivalent Boolean Expression for the following Logic Circuit.
The equivalent Boolean Expression for the given Logic Circuit is: F = (U’ + V).(V’ + W)
13(c). Write the SOP form of a Boolean function G, which is represented in a truth table as follows:
P Q R G
0 0 0 0
Ans.
0 0 1 0
0 1 0 1
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
The desired Canonical Sum-of-Product form is as following;
G = ∑(2, 3, 4, 6, 7) = P’QR’ + P’QR + PQ’R’ + PQR’ + PQR 13(d).
Ans.
Reduce the following Boolean expression using K-map: F(A, B, C, D) = ∑(3, 4, 5, 6, 7, 13, 15) There are 2 Pair and 1 Quad that reduce as given below:
Pair-1(m3+ m7) reduces to A’CD Pair-2(m4+ m7) reduces to A’BD’
Quad(m1+ m5+ m9+ m13) reduces to BD
Simplified Boolean expression for given K-map is F(A, B, C, D) = A’CD +’ A’BD’ + BD
14(a).
Ans.
Verify X’Y + XY’ + X’Y’ = (X’ + Y’) using truth table.
X Y X’ Y’ X’Y XY’ X’Y’ X’Y + XY’ + X’Y’ X’ + Y’
0 0 1 1 0 0 1 1 1
0 1 1 0 1 0 0 1 1
1 0 0 1 0 1 0 1 1
1 1 0 0 0 0 0 0 0
From Truth Table it is proved that X’Y + XY’ + X’Y’ = (X’ + Y’) 14(b).
Ans.
Write the equivalent Boolean Expression for the following Logic Circuit.
The equivalent Boolean Expression for the given Logic Circuit is: F = (X + Y’).(X’ + Z) 14(c).
Ans.
Write the POS form of a Boolean function H, which is represented in a truth table as follows:
A B C H
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 0
1 1 1 1
The desired Canonical Product-of-Sum form is as following;
H = π(0, 5, 6) = (A + B + C).(A’ + B + C’).(A’ + B’ + C) 14(d).
Ans.
Reduce the following Boolean expression using K-map:
F(P, Q, R, S) = ∑(1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 15)
There are 2 Pair and 1 Octet that reduce as given below:
Pair-1(m2+ m6) reduces to QR’S’
Pair-2(m4+ m12) reduces to P’RS’
Octet (m1+ m3+ m5+ m7 + m9+ m11+ m13+ m15) reduces to S Simplified Boolean expression for given K-map is
F(P, Q, R, S) = QR’S’ +’ P’RS’ + S 15(a).
Ans.
State and verify absorption law using truth table.
Absorption law states that (i) X + XY = X and (ii) X(X + Y) = X Truth Table for X + XY = X
X Y XY X+XY
Truth Table for X(X + Y) = X
X Y X +Y X(X + Y)