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00000CS 201121902

PART C
Answer all questions, each carries 3 marks.

Prove that the order of each sub group of a finite group G is a divisor of the
order of group G.
Let G be a group and suppose that a and b are any elements of G. Show if
(೩0) ‏“موك‎ then group is abelian.
Let (L,<) be 8 lattice and a,b,c,d elements of L.
Prove that if aDefine principle of duality in Boolean algebra.

PART D

Answer any two full questions, each carries 9 marks.

Prove that the necessary and sufficient condition that a non-empty subset of a
group © be a subgroup is a,b ६ प्र => ab! & H.
State and prove Absorption properties of lattice
Is ‏جردا‎ acomplemented lattice? Explain
[ 01215 set of divisors of 12 for the relation R={(x,y)|x divides y}]
Show that (7, *, *) is a ring where a*b=at+b-1 and a*b=a+b-ab
for every a, b € Z (Set of all integers).
Show that the set {1, 2, 3, 4, 5] is not a group under addition modulo 6
Define boolean algebra and explain how it is related to lattice.

PART 17

Answer any four full questions, each carries 10 marks.
Show that (9५५) ^ (p—r) A (6-೨1) tautologically implies ர without using truth

table

Express the following statements in predicate logic.
i)Some students are clever.

ii)All men are kind.

iii) Some person in this class has visited the Grand Canyon.

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