What does probability testing means? Describe how Fermat's Little Theorem tests for a prime number with suitable example. [5]

List any two applications of conditions of probability. You have 9 families you would like to incite to a wedding. Unfortunately, you can only invite 6 families. How many different sets of invitations could you write? [5]

Solve the recurrence relation $a_n = 5a_{n-1} - 6a_{n-2} $, with initial conditions $a_1 = 1$ and $a_2 = 2$. [5]

Prove that $5^{n}-1$ is divisible by 4 using mathematical induction. [5]

Consider a set U = {1,2,3,4,5,6,7,8,9,10}. What will be the computer representation for set containing the numbers which are multiple of 3 not exceeding 6? Describe injective, surjective and bijective function with examples. [10]

Compute the values. 3 mod 4 , 7 mod 5, -5 mod 3 , 11 mod 5 and -8 mod 6 Write down the recursive algorithm to find the value of $b^n$ and prove its correctness using induction. [5+5]

Find the values of x such that x = 1 (mod 5) and x = 2 (mod 7) using Chinese remainder theorem. [5]

Prove that the product xy is odd if and only if both x and y are odd integers. [5]

Let A = 'Aldo is Italian' and B = 'Bob is English'. Formalize the following sentences into proposition.a. Aldo isn't Italian.b. Aldo is Italian while Bob is English.c. If Aldo is Italian then Bob is not English.d. Aldo is Italian or if Aldo isn't Italian then Bob is English.e. Either Aldo is Italian and Bob is English, or neither Aldo is Italian nor is Bob English. [5]

What is S-D cut? For the following network flow find the maximal flow from S to D. [10]

Define Euler path and Hamilton path with examples. Draw the Hasse diagram for the divisibility relation on the set {1,2,5,8,16,32} and find the maximal, minimal, greatest and least element if exist. [5]

Define spanning tree and minimum spanning tree, mention the conditions for two graphs for being isomorphic with ean example [5]