Page 49 - Computer science 868 Class 12
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=A'(1 + B + C'.D) [Distributive law]
=A'.1 [Properties of 0 and 1]
=A'
(b) [(C.D)'+A] + A + C.D + A.B [ISC 2011]
=(C.D)' + C.D + A + A + A.B [Associative law]
=1 + A + A + A.B [De Morgan’s law]
=1 [Properties of 0 and 1]
(c) A.[B + C.(A.B + A.C)'] [ISC 2011]
=A.[B + C.( (A.B)'+(A.C)')] [De Morgan’s law]
=A.[B+C.( (A'+B').(A'+C'))] [De Morgan’s law]
=A.[B+ C.( A'+B'.C'))] [Distributive law]
=A. [ B + A'.C] [Complement law]
=A.B + A.A'.C [Distributive law]
=A.B [Complement law]
(d) A'.B + A.C + B.C
=A'.B + A.C + B.C.1 [Properties of 0 and 1]
=A'.B + A.C + B.C.(A+A') [Complement law]
=A'.B + A.C + A.B.C + A'.B.C [Distributive law]
=A'.B + A'.B.C + A.C + A.B.C [Associative law]
=A'B(1+C) + A.C(1+C) [Distributive law]
=A'.B + A.C [Properties of 0 and1]
(e) A.B' + A'.B.C' + (A.C)' + B.C
=A.B' + A'.B.C' + A' + C' + B.C [De Morgan’s law]
=A.B' + A'.(B.C'+1) + (B+C').(C+C') [Distributive law]
=A.B' + A'+ B + C' [Complement law]
=(A+A').(A'+B') + B + C' [Distributive law]
=A' + B' + B + C' [Complement law]
=A' + 1 + C' [Complement law]
=1 [Properties of 0 and 1]
7. Write the cardinal form of the following canonical expressions:
(a) A.B.C'.D + A.B'.C'.D + A'.B.C'.D + A.B.C.D' + A.B.C.D + A'.B'.C'D + A.B'.C'.D' [eliminating duplicates]
Ans. F(A, B, C, D) = Σ(1101, 1001, 0101, 1110, 1111, 1000)
=Σ(13, 9, 5, 14, 15, 1, 8)
=Σ(1,5,8,9,13,14,15)
(b) (A+B+C+D).(A'+B'+C'+D).(A+B'+C'+D').(A'+B+C'+D).(A+B'+C+D')
Ans. F(A, B, C, D) = π(0000, 1110, 0111, 1010, 0101)
=π(0, 14, 7, 10, 5)
=π(0, 5, 7, 10, 14)
8. Express X'+Y in canonical SOP.
Ans. X'+Y
=X'.1 + Y.1
=X'.(Y+Y') + Y.(X+X')
=X'.Y + X'.Y' + X.Y + X'.Y
=X'.Y + X'.Y' + X.Y
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Boolean Algebra 47

