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R3912 Pages:

positive integer for which 8" = e. (೮15 the identity element)

Show that every finite integral domain is a field.

Let (L, *, ®) be any distributive lattice. For any a, b, c element of L, show that

(a*b = a*c) and (8029 5 80) ५) ==> 9 = ५

Define a Boolean algebra. Illustrate a two element Boolean Algebra with an
example

Prove that every finite group of order n is isomorphic to a permutation group of
degree n.

Define Lattice homomorphism and direct product of two lattices.

PART E
Answer any four full questions, each carries 10 marks.

Without using truth tables prove that 17 ^ ©) ೨ (1? ५ (IP ۷ Q)) © (1? ५ 0)

Using truth table determine whether the conclusion 1P follows logically from the
premises P—Q and 1(P ^ 0).

Show that Rv ऽ follows logically from the premises C V D, (C V D) > 1H,

1प-> (A ^+ 1B), and (A ^ 1B) ೨ (RV S).

If there was a ball game, then travelling was difficult. If they arrived on time,
then travelling was not difficult. They arrived on time. Therefore, there was no
ball game. Show that these statements constitute a valid argument.

Symbolise the statement “x is the father of the mother of y”

Indicate the variables that are free and bound

1. (x) (P(x) പ്രോ AR(x)

11. (x)(P(X) A (30) Q(X) വ്ര)

Use mathematical induction to prove that 73 + 2n 15 8 multiple of 3 for all n> 1.
Show that

(Ax)(P(x) A O(x)) => (Ax) P(x) க (Ax)O(x)

Test the consistency of the following statements.

“Tf Jack misses many shootings, then his films will be a flop. If Jack’s film flops,
then he fails to act. If he signs for many films, then he does not fail to act.” Jack
misses many shootings and signs for many films.

Show that R-S can be derived from P—(Q-—S), RvP and Q

Using proof by contrapositive method, prove that “if 31 + 2 is odd, then n is odd”
Use mathematical induction to prove that 12 + 22 + 37+ ... + 175 (n/6)(n +1)(2n
+1) for 8111 EN.

اد بد عاد 3

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