a) Explain the following:

i) Euler Graph

ii) Isomorphic graphs

iii) Minimal spanning tree

iv) Height of the tree (Unit 4)

b) Let $Z$ be the group of integers with binary operation $*$ defined by $a * b = a + b - 2$, for all $a, b \in Z$. Find the identity element of the group $\langle Z, * \rangle$. (Unit 2)

a) Prove that the Complement of each element in a Boolean Algebra B is unique. (Unit 5)

b) Let $A$ be any finite set and $P(A)$ be the power set of $A$. $\subseteq$ be the inclusion relation on the elements of $P(A)$. Draw the Hasse diagrams of $(P(A), \subseteq)$ for the following:

i) $A = \{a\}$

ii) $A = \{a, b\}$

iii) $A = \{a, b, c\}$

iv) $A = \{a, b, c, d\}$ (Unit 5)

a) i) Prove that $p \land q \Rightarrow q \lor p$ is a Tautology.

ii) Show that $(p \lor q) \land (\neg p) \land (\neg q)$ is a contradiction. (Unit 3)

b) Explain complete digraph and Euler Graph using suitable example of both. (Unit 4)

a) Define planar graph. Prove that for any connected planar graph, $v - e + r = 2$ where $v, e, r$ is the number of vertices, edges, and regions of the graph respectively. (Unit 4)

b) Prove that the relation $R$ defined by "a is congruent to b modulo m" on the set of integers is an equivalence relation. (Unit 1)

a) Prove that $5^{2n} - 1$ is divisible by 24, where $n$ is any positive integer. (Unit 1)

b) Draw the Hasse diagram representing the positive divisors of 36 and 45. (Unit 5)

a) Show that the relation 'R' defined by $(a, b) R (c, d)$ iff $a + d = b + c$ is an equivalence relation. (Unit 1)

b) Explain various Rules of Inference for Propositional Logic. (Unit 3)

a) Show that every Cyclic group is Abelian. Prove that a lattice with 5 elements is not a Boolean algebra. (Unit 2 / Unit 5)

b) Define Pigeon hole Principle. Write the contra positive of the implication: "if it is Sunday then it is a holiday." (Unit 1 / Unit 3)

a) Prove that $G = \{0, 1, 2, 3, 4, 5, 6\}$ is an abelian group of order 7 with respect to addition modulo 7. (Unit 2)

b) Prove or disprove that intersection of two normal subgroups of a group G is again a normal subgroup of G. Define subgroup, normal subgroup, Quotient group, with an example for each. (Unit 2)