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00000CS201121901

PART ட்‌
Answer all questions, each carries 3 marks.
Show that the set {0,1,2,3} is not a group under multiplication modulo 4.
Define ring homomorphism.
Draw the lattice for ഗാം. /> where [230 be the set of all divisors of 30. / denotes
the relation divides.
Explain principle of duality in Boolean algebra.
PART D
Answer any two full questions, each carries9 marks.
If * is the operation defined on 5 =© > 0, where © is the set of rational
numbers and * is given by (a,b) * (c,d)=(ac,bc+d). Find whether (S,*) is a
group?
Let < 020, |> denote the poset of all divisors of 20.
Show that [220 is a lattice.
Explain Distributive lattice with an example.
Show that (7, 0, ಅ) is a ring where a 0 9 = atb-1 and a © b = at+b—ab
Prove that the order of a subgroup of a finite group divides the order of the
group.
Simplify the boolean algebraic expression AB+A (B+C)+B(B+C).
PARTE
Answer any four full questions, each carries 10 marks.

Show that (t A s) can be derived from premises p> ‏و ,و‎ > | 1, 1, pV(t A ൭).
Symbolize the following statement. (i). All men are giants. (ii). Given

any positive integer there is a greater positive integer.

Show that the following premises are in consistent.

If Ram gets his degree he will go for a job. If he goes for a job he will get
married soon. If he goes for higher study he will not get married. Ram gets
his degree and he goes for higher study.

Prove by contra positive that if 0 is even integer then n is even.

Show that (a—b) A(a—c), | (೧/೦), (dV a) 4

Show that from (4x)(F(x) A S(x)) — Vy(M(y)—> ५४(४))
(Ay)(M(y) ^ 1910)
Concludes (x) (F(x) ೫ S(x)).

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