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B3809 Pages: 2

of all divisors of n and D denote the relation ‘divides’.
Define sub Boolean algebra. Give one example.
PART D

Answer any two full questions, each carries 9 marks
Show that the set{ 0, 1, 2, 3, 4, 5) under addition and multiplication modulo 6 is
group or not.
Find all the subgroups of > 212, +12>
Define ring and field. Give one example to each.

A= {2, 3, 4, 6, 12, 18, 24, 36} with partial order of divisibility. Determine
whether the POSET is a lattice or not.
Show that the lattice < Sn, D > for n = 216 is isomorphic to the direct product of
lattices n = 8 and 1 = 27.
Define complemented lattice and distributive lattice. Give one example to each.
PART E
Answer any four full questions, each carries 10 marks
Prove that ‏م‎ v (६१1) and ( ‏م‎ v q)(p vr) are logically equivalent
Show that (ഇ (റു ^ r)) ए (६१1) ए ‏م)‎ ? 1) >-< 1
Show that 5 v ris tautologically implied by ‏م)‎ v q) ^ ‏و) ^ )1 > م)‎ -< 5(
Show that r ^ ( v q) is a valid conclusion from the premises p v q, ‏و‎ -< 1,
p->m, and ~m
“Tf there are meeting, then traveling was difficult. If they arrived on time, then
traveling was not difficult. They arrived on time. There was no meeting”. Show
that the statements constitute a valid argument.
Construct truth table for ~ (p ^ q) <> (~p ٢ ~प). Determine whether it is
tautology or not.
Show that (x) ( P(x) -> Q(x) ) ^ ७) (Q(x) > R(x) => (x) (P(X) -> R(x) )
Prove that (4x) ( P(x) ^ Q(x) ) => (Ax) P(x) ^ (4x) Q(x)
Symbolize the statements:
i) All the world loves a loverii) All men are giants.
Show that (4x) M(x) follows logically from the premises (x) (H(x) -> M(x)) and
(Ax) H(x)
Prove by contradiction that if > is an even integer then n is even.

Prove that 23" - | is divisible by 11 for all positive integers ൩.
‏عاد بد‎ ಶೇತೇ

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