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B7070

PART D
Answer any two full questions, each carries 9 marks.
Show that the order of a subgroup of a finite group divides the order of the group.

Define ring homomorphism.

Show that (13 ೦,೮) is acommutative ring with identity, where the operations © and © are
defined, for any ೩0೮1 as 9 © b=a+b-1 anda © b= atb-ab.
Let (L,S) be a lattice and a,b,c,de L. Prove that if a > ‏ء‎ and b< 6, then
(1) aVb(ii) aAbShow that in a Boolean algebra,for any a,b,c

(4 ^ ¢ ^+ ८) ८ (® ^ ८) = ० ^ ८
PART E
Answer any four full questions, each carries 10 marks.
a) Construct truth table for(~pA(~qAr))V((qAr)V(pAr))
Explain proof by Contrapositive with example.

Prove the following implication

(x)(P(x) VQ(x))==> (x) P(x) ^ (Ax) Q(x)

Represent the following sentences in predicate logic using quantifiers
(i) “x is the father of the mother of y”
(ii) “Everybody loves a lover”

Determine whether the conclusion C follows logically from the premises
प्रा: ~pVq, Ho: ~(qA~r), ‏:متا‎ - ಲಿ: ~p

Without using truth table prove p > (q > p) <=> ~p> ( > ५)

Determine the validity of the following statements using rule CP.
“my father praises me only if I can be proud of myself. Either I do well in sports or I

can’t be proud of myself. If I study hard, then I can’t do well in sports. Therefore if my
father praises me then I do not study well”

Show that r > s can be derived from the premises p >(q >s), ‏ہہ‎ V p, 4

Prove, by Mathematical Induction, that n(n + 1)(n + 2)(n + 3) is divisible by 24, for all
natural numbers n

“If there are meeting, then travelling was difficult. If they arrived on time, then
travelling was not difficult. They arrived on time. There was no meeting”. Show that
these statements constitute a valid argument.

Show that 2" > n! For n> 4

اد ‎೫ ೫‏ اد

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