a) Define various types of functions. How many symmetric and reflexive relations are possible from a set A containing ‘n’ elements? (Unit 1)

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) 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) Show that there does not exist a graph with 5 vertices with degrees 1, 3, 4, 2, 3 respectively. (Unit 4)

b) Obtain the generating function for the sequence 4, 4, 4, 4, 4, 4. Explain complete digraph. (Unit 5 / 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) Find the numbers between 1 to 500 that are not divisible by any of the integers 2 or 3 or 5 or 7. (Unit 5)

a) A collection of 10 electric bulbs contain 3 defective ones. (i) In how many ways can a sample of four bulbs be selected? (ii) In how many ways can a sample of 4 bulbs be selected which contain 2 good bulbs and 2 defective ones? (iii) In how many ways can a sample of 4 bulbs be selected so that either the sample contains 3 good ones and 1 defectives ones or 1 good and 3 defectives ones? (Unit 5)

b) Discuss about Complete digraph and Euler Graph with the help of suitable examples. (Unit 4)

a) Solve the following recurrence equation using generating function.
$G(K) - 7 G(K-1) + 10 G(K-2) = 8K + 6$. (Unit 5)

b) Prove the validity of the following argument "if the races are fixed so the casinos are crooked, then the tourist trade will decline. If the tourist trade decreases, then the police will be happy. The police force is never happy. Therefore, the races are not fixed." (Unit 3)

a) Let $(L, \vee, \wedge, \leq)$ be a distributive lattice and $a, b \in L$. if $a \wedge b = a \wedge c$ and $a \vee b = a \vee c$ then show that $b = c$. (Unit 5)

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

a) What is Ring? Define elementary properties of Ring with example. (Unit 2)

b) Prove or disprove that intersection of two normal subgroups of a group G is again a normal subgroup of G. (Unit 2)