Explain fuzzy set with example. How do you find complement of a fuzzy set? [5]

What is product rule? How many strings are there of four lowercase letters that have the letter x in them? [5]

How can you use mathematical induction to prove statements? Use mathematical induction to show that sum of first n positive integer is $\frac{n(n+1)}{2}$ [10]

Explain linear homogeneous recurrence relation with constant coefficients. What is the solution of the recurrence relation $a_n = 6a_{n-1} - 9a_{n-2}$, with initial conditions $a_0 = 1$ and $a_1 = 6$? [10]

Define function. Let $f_1$ and $f_2$ be function from R to R such that $f_1(x) = x^2$ and $f_2(x) = x - x^2$. What are the functions $f_1 + f_2$ and $f_1 * f_2$? [5]

What is congruent modulo? Determine whether 37 is congruent to 3 modulo 7 and whether -29 is congruent to 5 modulo 17. [5]

Give an example of tautology and contradiction. Show that implication and contrapositive are equivalence. [5]

What is direct proof? Give a direct proof that if m and n are both perfect squares, then mn is also a perfect square. [5]

What is shortest path problem? Use Dijkstra's shortest path algorithm to find the shortest path between vertices a and z in the weighted graph below: [10]

Let us assume that R be a relation on the set of ordered pair of positive integers such that $((a, b), (c, d)) \in R$ if and only if ad = bc. Is R an equivalence relation? [5]

Define network flow with example. What are saturated edge, unsaturated edge and slack value? [5]

Explain the matrix representation of relations with example. [5]