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C192006 Pages:3

(c,d) iffatd = b+c

(i) Prove that R is an equivalence relation
(ii) Find the equivalence class of (2,5)
PART C

Answer all questions, each carries3 marks.
Show that the order of a subgroup of a finite group divides the order of the
group.
Prove that the set consisting of the fourth roots of unity forms an abelian
group with respect to multiplication composition.

In a distributive lattice 8 ۷ 0 = 8 ५ cand a\b=aAcimplythatb=c.
Show that the complement of every element in a boolean algebra is unique.

PART D
Answer any two full questions, each carries9 marks.
The necessary and sufficient condition that a non-empty subset H of a Group
© bea subgroup is a€H, b€ H> ab '€H
Let ೫, y be arbitrary elements in a boolean algebra (3, +, . ,' ,0, 1). Prove the
De-Morgan’s Law (x+y)'= x'y'.

Show that the lattice with three or fewer elements is a chain.

What is Ring with Unity? Give an example of a commutative ring without

unity.

If the order of a group G be 'n' ie a"=e then the set H = { a,a” ....,a"} forms a
group with respect to the multiplication composition in G
Define a bounded lattice. Give an example.

PART E

Answer any four full questions, each carries10 marks.
Show the following implication without constructing the truth table
(PY ൯൨൭൭൭ (७ VIP} >R) = (Q>R)

Negate the following statements and give the logical expression

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