III BA, B.Sc. Mathematics Paper IV (Elective-2) - Acharya Nagarjuna ...

III B.A./B.Sc. Mathematics8Paper IV (Elective -2) - Curriculum58. Two one-dimensional arrays X and Y each contain 50 elements. Write the ALGOL program to computeLX=50Σj=1X2j(Length of the vector X ), LY =50Σ Yj=12j(Length of the vector Y )INPROD =50Σj=1XYjj(Inner product of X and Y )⎡a a a59. Write ALGOL program to multiply the matrix A =⎢a a a⎢⎣⎢a a a11 12 1321 22 2331 32 3360. Write down the ALGOL block for finding the roots of ax 2 + bx + c = 0 ( a ≠ 0)⎤⎡b1⎤⎥⎥ by a column matrix B =⎢b⎥⎢2⎥⎦⎥⎣⎢b3⎦⎥, .UNIT - 4 (BOOLEAN ALGEBRAS)61. What is two element Boolean algebra.62. In a Boolean algebra show that x≤ z⇒ x∨( y∧ z) = ( x∨ y) ∧z.63. In a Boolean algebra prove that ( x∧ y′ ) ∨ ( x′ ∧ y) = ( x∨ y) ∧ ( x′ ∨ y′ ).64. In a Boolean algebra prove that ( x∧ y) ∨( y∧z) ∨( z∧ x) = ( x∨ y) ∧( y∨z) ∧( z∨ x).65. Let ab , ∈ B,a Boolean algebra. If ∨ is denoted by + then prove that a+ b is an upper bound for theset { ab , } and also a+ b=sup{ a,}.b66. Given the interval [ ab , ] of a Boolean algebra A . Show that the algebraic system [[ ab , ], ∧∨∗ , , , ab . ] isa Boolean algebra, where x ∗ = ( a∨ x′ ) ∧b, ∀x∈[ a, b].67. Draw the block diagram of p∧( p∨q).68. Draw the block diagram of ( A ∧ A ) ∨ ( A′ ∧ A′)1 2 1 269. Draw the block diagram of ABC ∨ A′ B′ C ∨ A′ B′ C′.70. Write the gating network representing the Boolean expression ( x∨ y) ∧ ( x′ ∨ y′ ∨ z′ ) ∧ ( y′ ∨ z).71. Write the gating network representing the Boolean expression [( x ∨x ) ∧ x′ ] ∨( x ∧ x ) .72. Write the Boolean expression for the gating network.xy1 2 3 1 2z73. Show that [( p⇒q) ∧( p⇒r)] ⇒( p⇒ r)is a tautology.74. Show that ( p→q) →[( r∨ p) →( r∨ q)]is tautology, regardless of r .