B.A. / B.Sc. Maths Semester-5th Previous Year Old Question Paper Jammu University, Under Graduation

B.A. / B.Sc. V Semester Examination

BS-VS/11

229006

MATHEMATICS

Course NO.: 501

Time Allowed: 3 hours                                                                       Maximum Marks: 80

Note: Attempt any five questions, selecting one from each unit. All questions carry equal marks.

Unit – I

1. a) Define the Cartesian product of two sets A and B. Show that (A×B) ∩ (C×D) = (A∩B) × (B∩D) for any sets A, B, C, and D.

b) Define a relation and inverse of a relation. If R^-1 and S^-1 are inverse of relations R and S, respectively, then show that (SoR)^-1 = R^-1oS^-1.

2. a) If R is an equivalence relation on a set X and [ɑ] denotes the equivalence class of ɑ ∈ X, the show that [ɑ] = [b] if and only if ɑ ∈ [b], ∀ ɑ, b ∈ X.

b) If ƒ: A ➝ B and g: B ➝ C are maps such that goƒ: A ➝ C is onto, then show that g is onto but ƒ need not be onto.

Unit – II

3. a) Give an example of the following:

i) A finite non-abelian group.

ii) An infinite non-abelian group.

b) Define the order of a group. If G is a group of even order, then prove that there is an element ɑ ≠ e in G such that ɑ² = e.

4. a) Define a subgroup of a group. Let G be a group and H = {a ∈ G; ab = ba, ∀ b ∈ G}. Prove that H is a subgroup of G.

b) Define a cyclic group. Prove that every group of prime order is cyclic.

Unit – III

5. a) Define left coset of a subgroup in a group. Prove that any two left cosets of a subgroup H in a group G are either identical or disjoint.

b) If H and K are subgroups of a group G, then prove that HK is a subgroup of G iff HK = KH.

6. a) Give an example of a non-abelian group G a normal subgroup H of G such that the quotient group G/H is abelian.

b) Prove that every proper subgroup of an infinite cyclic group is infinite.

Unit – IV

7. a) Let G be a group and ɑ be any fixed element of G. Prove that the mapping ƒ: G ➝ G defined by ƒ(x) = xɑx^-1, ∀ x ∈ G is an isomorphism of G onto G.

b) Show that any cyclic group of order n is homomorphic to the additive group of residue classes modulo n.

Unit - V

9. a) Give an example of a ring which contains elements a, b such that (a + b)² ≠ a²  + 2ab + b².

b) Show that a finite integral domain is a field.

10. a) Define left ideal of a ring. Show that any intersection of left ideals of a ring is also a left ideal, but the union of two left ideals need not be a left ideal.

b) Show that an ideal M of a commutative ring R with unity is a maximal ideal iff R/M is a field.

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