First Question of compulsory.

Answer any FOUR from the remaining questions.

All questions carry wqual marks.

Answer All Parts of any questionj at one place.

1. (a) Let A and B be sets. Show that

(i)                  (AB)A

(ii)                A(AB).

(b) Use mathematical induction to prove that n3-n is divisible by 3 whenever nis a positive integer.

(c) What is the coefficient of x’2y’3 in the expansion of (x+y)25?

(d)  Define Non-Homogeneous recurrence relation.

(e)  State Four color theorem.

(f) Prove that is G is a tree, then the sum of degrees equals |v|-2.

(g) Show that a vertex v is a tree T if a cut vertex of T iff deg(v)>1.

2. (a)  Construct the truth table for

[(pvq)^(~r)]-><-(q->r)

(b) Verify that the following argument is valid by translating into symbols and using truth tables to check for tautologies.

If Joe is a mathematician, then he is ambitious.

If Joe is an early riser, the he does not like out meal

If Joe is ambitiors, thenhe is an early riser.

Hence, if Joe is a mathematician, then he does not like out meal.

3.(a)   Prove that the function b(n)=2(3n)-5 is the unique function defined by

(i)                  B(0)=-3, b(1)=1 and

(ii)                B(n)=4b(n-1) -3b(n-2) for n>=2.

(b) Solve the recurrence relation

an-7an-1+10an-2=0 for n>=2.

4.(a) Let R1 and R2 be two equibvalence relations on asset A. Show that R1 R2 is an equivalence relations on a set A but that R1R2 beed not be an equivalence relation.

(b) If Ris a symmetric relation defined on a set A, then prove that RR2 is symmetric.

5.(a)  Let Rbe a relation on A=(a,b,c,d) whose adjacency martix is given              Computes the adjacency matrix of R+ using Warshall’s Algorithm.

(b) Draw a digraph with 5 verteces that has 4 strongly connected components, 2 weekly connected components and 3 unilaterally connected componints oxactly.

6. (a) Prove that a simple non-directed graph Gis a tree iff G is connected and contains no cycles.

(b)  Draw all regular binary trees:

(i) With exactly  7 vertices.

(ii) with exactly 9 vertices.

7.Show that every subsystem of Boolean algebra is a Boolean Algebra.

8.Write short notes on the following:

(a)    Language and grammar.

(b)   Finite State machine.

(c)    Turning machine.

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First Question of compulsory. Answer any FOUR from the remaining questions. All questions carry wqual marks.  Answer All Parts of any questionj at one place. 1. (a) Let A and B be sets. Show that (i)                  (AB)A (ii)                A(AB). (b) Use mathematical induction to prove that n3-n is divisible by 3 whenever nis a positive integer. (c)...