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02000CS201092002

PART D

Answer any two full questions, each carries 9 marks.
What is a complete lattice? Give an example.
Show that the set of all positive rational numbers Q+ forms an abelian group
under the operation * defined by a*b=(ab)/2 for a,b € 0+.
Define a Boolean algebra. Illustrate a two element Boolean Algebra with an
example.
Let H = {0, 3, 6} in 29 under addition. What are the cosets of H in 77

Verify that the set{ 0, 1, 2, 3, 4, 5 } under addition and multiplication modulo 6
is group or not.
A= {2, 3, 4, 6, 12, 18, 24, 36} with partial order of divisibility. Determine
whether the POSET is a lattice or not.
PARTE
Answer any four full questions, each carries 10 marks.
Without using truth tables prove that ۱)۲ ^ Q) > (1? ५ (1? ५ Q)) <=> (IP ५ Q)

Suppose x is a real number. Consider the statement "If x 2 = 4, then x = 2."
Construct the converse, inverse, and contrapositive.
Prove that ‏م‎ v (q Ar) and ( 9 ४ q) A (01 1) are logically equivalent.

Prove that (4x) ( P(x) A Q(x) ) => (35) P(x) A (Ax) Q(x).

Show that the premises ^“ A student in this class has not read the book” and
“ Everyone in this class passed the first exam ” imply the conclusion
“Someone who passed the first exam has not read the 90007,

Show that the premises, “ 1! is not sunny this afternoon and it is colder than
yesterday” , “ We will go swimming only if it is sunny” , “ If we do not go
swimming , then we will take a canoe trip” , and “ If we take a canoe trip , then
we will be home by 5011501 '' lead to the conclusion “ We will be home by
sunset”.

Show that (x) ( P(x) -> Q(x) ) A (x) (000) -> R(x) => (x) (P(X) -> R (x) )

Show that R A (P ५ Q) is a valid conclusion from the premises 1௫0,027,
? -> ۷ 810 ¬ ७.

Show that (4x) M(x) follows logically from the premises (x) (H(x) -> M(x))
and (4x) H(x)

Show that ((P > 0) ^ (ಛಿ R)) - (P— 1२) is a tautology.

Prove by contradiction “If 3n + 2 is an odd integer , then n is odd “.

Show that ऽ ۷ R 15 tautologically implied by (P ५ Q)A(P —R ) A(Q—S)

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