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DM-2022

Q-1

(a) Using mathematical inductions ,prove that n^3+2n is divisible by 3.

(b) Prove that (A ∪ B)’=A’ ∩B’

Q-2

Construct the truth table for the following statements : (i) – (pɅq)Ʌ (-r). (ii)- (pɅ-q) v (r).

Q-3

(a) R = {(a, b) | length of string a = length of string b} on the set of strings of English letters. Prove that R is an equivalence relation.

(b) Show that the inclusion relation is a partial ordering relation on the power set of a set.

Q-4

(a) Given A = {1, 2, 3), B = (a, b) and C = (1, m, n). Find each of the following sets (i) AxBX C (ii) A x C. (iii) B XCx A.

(b) Define Lattice. Prove that D36 the set of divisors of 36 ordered by divisibility forms a lattice.

Q-5

(a) Prove that the function f: NN defined as N→N defined as f(n) = [n+1, nis odd n-1, nis even is inverse of itself.

(b) Solve: an + an-1 function method. = 3n2″, ao = 0, using Generating

Q-6

Q-7

(a) Prove that the identity element in a group is unique.

(b) Let G be a group and a € G. Prove that the cyclic subgroup H of G generated by a is a normal subgroup of N(a) = {xe G: xa = ax).

Q-8

(a) Let P be a subgroup of a group G and let Q = {xe G: xP = Px}.

Is Q a subgroup of G?

(b) Let f: (R,+) → (R,, x) is defined s f(x) = ex for all x in R, where R→ set of real numbers ond R. → set of positive real numbers. Prove that ƒ is a homomorphism. Is f an isomorphism?


DM-2021

Q-1

(a) Show that 1²+3² +5² + …………+ (2n- 1)²= n(2n-1) (2n+1)/3 by mathematical induction.

(b) Given that (A ∩ C) ⊆ (B ∩ C) (A ∩ C) ⊆ (B ∩ C) show that A ⊆ B.

Q-2

(a) Construct the truth tables for the following statements (i) (p→p) → (p→-p), (ii) (pv-q) → -p. (iii) p↔ (-pv-q).

(b) Let A, B and C be sets such that (A∩ B ∩ C) = ∅ , (A ∩ B) ≠ ∅ , (A ∩ C) ≠ ∅ and (B ∩ C) ≠∅ . Draw the corresponding Venn diagram.

Q-3

(a) Find all the partitions of B = {a, b, c, d}.

(b) Let A = {a, b} and B = {4, 5, 6}. Given each of the following : (i) A × B (ii) B × A (iii) A × A (iv) B × B.

Q-4

(a) Show that if R1 and R2 are equivalence relations on A, then R1 R2 is an equivalence relation on A.

(b) Let A = Z, the set of integers and let R is defined by a R b if and only if a ≤ b. Is R is an equivalence relation ?

Q-5

(a) Prove that if f : A -> B and g : B -> C are one-to-one functions, then gof is one-to-one.

(b) Let A = B = C = R, and let f : A -> B and g : B -> C be defined by f(a) = a – 1 and g(b) = b² find : (i) (fog) (2) (ii) (gof) (2) (iii) (fof) (y) (iv) (gog) (y).

Q-6

(a) Let A and B be two finite set with same number of elements, and let f : A  B be an everywhere defined functions : (i) If f is one-to-one, then f is onto. (ii) If f is onto, then f is one-to-one.

(i) If f is one-to-one, then f is onto.

(ii) If f is onto, then f is one-to-one.

(b) If n pigeons are assigned to m pigeonholes, and m < n, then atleast one pigeonhole contains two or more pigeons.

Q-7

(a) Define the following :

(i) Group.

(ii) Cyclic group.

(b) Let H and K be subgroups of group G :

(i) Prove that H ∪K is subgroup of G.

(ii) Show that H∩K need not be subgroup of G

Q-8

(a)

(b) Let G be a group, and let H = {x/x belongs to G and ax = xa for all a belongs to G}. Show that H is a normal subgroup of G

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