### For Answers/Solution Just Click on the Questions.

## 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

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

## Q-4

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

## Q-5

(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.

## 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?

## 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

## Q-3

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

## Q-4

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

## Q-5

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

## 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.

## Q-7

(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)