ASSIGNMENT PROBLEMS
UNIT-I
1) Without using truth table show that (PQ)  (P (PQ))  (PQ)
2)Without using truth table obtain the product of sums canonical form of the formula
(PR)  (Q↔P). Hence find the sums of product canonical form.
3) Show that the following premises are inconsistent. If the contract is valid then John
is liable for penalty.If John is liable for penalty then he will go bankrupt. If the bank
will loan him money, he will not go bankrupt. As a matter of the fact, if the contract is
valid and the bank will loan him money.
4) Show that the following premises are inconsistent.
i) If Jack misses many classes through illness, then he fails high school.
ii) If Jack fails high school, then he is uneducated.
iii) If Jack reads a lot of books, then he is not uneducated.
iv) Jack misses many classes through illness and reads a lot of books
5)Prove that 2 is irrational by giving a proof using contradiction
UNIT-II
6)Prove by mathematical induction, that 𝟏 + 𝟐 + 𝟑 + ⋯ . . +𝒏 =
𝒏(𝒏+𝟏)
𝟐
7)State and Prove Pigeonhole principle with example.
8) Find an explicit formula for the Fibonnaci sequence.
9) Determine the number of positive integers n, 1≤n≤1000 that are not divisible by 2,3 or 5
but are divisible by 7.
10)Solve the recurrence relation 𝒂𝒏 − 𝟑𝒂𝒏−𝟏 = 𝒏; 𝒂𝟎 = 𝟏 using generating functions.
UNIT-III
11)State and prove Handshaking Theorem.
12) Prove that a simple graph with n vertices be connected if it has more than
(𝒏−𝟏)(𝒏−𝟐)
𝟐
.
13) A connected graph is Euler graph (contains Eulerian circuit) iff it each of its
vertices is of even degree.
14) Let G be a simple undirected graph with n vertices. Let u and v be two non adjacent
vertices deg(u) +deg(v) ≥ n in G. S.T G is Hamiltonian iff G+uv is Hamiltonian.
15) Show that a graph is disconnected iff vertex set V can be partitioned into two non
empty subset V1 and V2 such that there exists no edge in G whose one end vertex is in V1
and the other in V2.
UNIT-IV
16)  on R defined by x  y = x + y + 2xy  x, y  R. Check (i) (R, ) is a monoid or not;
(ii) Is it commutative; (iii) Which elements have inverses and what are they?
17) Let S = Q X Q, be the set of all ordered pairs of rational numbers and given by
(a, b)  (x, y) = (ax, ay+b). (i) Check (S, ) is a semigroup. Is it commutative?.
(ii) Also find the identity element of S.
18) Prove that a group (G,) is abelian if and only if (a  b)2 = a2  b2.
19)State and prove Fundamental theorem of group homomorphism.
20) P.T the set Z4=(0,1,2,3) is a commutative ring w.r.to the binary operation +4 & x4
UNIT-V
21)Draw the Hasse diagram for P1={2,3,6,12,24,36} & P2={1,2,3,4,6,12} and ≤ is a relation
such that x≤y iff x/y. 1) Find LUB and GLB for (P2, R) 2) Find LUB and GLB for (2,3) &
(24,36) in P1
22) State and prove distributive inequality of Lattice
23) If S42 is the set of all divisors of 42 & D is the relation on S 42, prove that { S42, D} is a
complemented lattice
24)In any Boolean algebra show that a=b iff 𝒂𝒃 + 𝒂𝒃 = 𝟎
25) Simplify the Boolean expression a’.b’.c + a.b’.c + a’.b’.c’