mfcs mathematics

8/13/2019 MFCS mathematics

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SUB: MFCS FACULTY: S. JAGADESH BABU

UNIT 1:

1) (a) Determine the truth value of each of the following statements

i. 6 + 2 = 7 and 4 + 4 = 8.ii. four is even.iii. 4 + 3 = 7 and 6 + 2 = 8.

(b) Write each of the following statements in symbolic formi. Anil & Sunil are rich.

ii. Neither Ramu nor Raju is poor.iii. It is not true that Ravi & Raju are both rich.

(c) Write a short note on normal forms

2) (a) Prove that (P Q )V R ((P V Q) R) is a con tradiction. (b) Obtain PDNF and PCNF of the following formula (PVQ) (P Q)

3) (a) S h o w the following implication without cons tru ct ing th e truth table.i. (P Q) Q (PV Q)

ii. P Q P (PQ) (b) Show that the proposition PQ and (PVQ) (P) (Q) are con tradiction

4) (a) If P is true, Q is false and R is true, then find the truth value of((P Q) R )V (P V R) without constructing the truth table.

(b) Find the disjunctive normal form of P V (P (QV (Q R))).( c) Show that the following statements is a tautolog y.

((PVQ) (P R) (Q R)) R

UNIT 2:1) (a) Show that R (PVQ) is a valid conclusion from premises PVQ,Q R, P M andM

(b) Explain the use of predicates with suitable examples2). (a) S ho w that (x) (P (x) Q (x)) (x) (Q (x) R (x)) (x) (P (x) R (x))

Using rules of inference.

(b) Show that ( x) M(x) follows logically from the premises x (H (x) M (x)) &( x)H (x)

3) (a) Prove or disprove the validity of the following argumen ts. No mathematicians are ignoran t.All ignoran t people are haugh t y.

Hence, some haugh t y people are not mathematicians.

(b) Prove the following logical x [p (x) Q (x)] x, (p (x) x, Q (x)

4) (a) Show that x(P (x)Q(x)) x(P (x)xQ (x)) (b) Negate each of the following expre ssions

i. (x) (y) R (x, y) ii. (x) (y) R (x, y)

iii. (xy R (x, y) xyP (x, y (c) Explain the term quantifier.


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UNIT 3

1. (a) A function f(ZZ) Z is defined by f(x,y) = 4x = 5y. Prove that f is not One-to-one, but onto(b) If A, B, C are three sets such that A B. Show that (A C) (B C)(c) If A = {1, 2, 3}, B = {4, 5}. Findi. ABii. BA

2. (a) Let f(x): x 2-3x+2. Findi. f(x 2)ii. f(x+3)(b) Prove that A (B C) = (A B) (A C) (c) Define equivalence relation

3.(a) Let L be a Lattice. Then prove that the relation a b defined by either ab = a (or)aVb = b is a partial ordering relation on L.

(b) Let X = {1,2,3,4 } . Define a function f: x x. Such that f = Ix and isOne-to-one. Find f 2, f 3 , f 1 , f of 1

4.(a) Suppose R is an anti-Symmetric relation of a set A. Show that R S. R 1 isanti-symmetric, for any relation on A.

(b) Let f: R R & g: R R. Where R is a set of real numbers find fog and gof, wheref(x) = x2 -2, g(x) = x+4. State whet her these functions are injectiv e, surjective or bijective.


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UNIT 4:

1. (a) If G = , H = and K =


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1. Suppose that Florida State University has a residence hall that has 5 single rooms, 5

double rooms, and 3 rooms for 3 students each. In how many ways can 24 studen ts be assigned to the 13 rooms?

2. Six new employees, t wo of whom are married to each other are to be assigned sixdesks that a r e lined up in a row. If the assignmen t of employees to desks is maderandomly, wh at is the probability t h a t t h e married couple will ha ve non-adjacen t desks?

3. (a) Find the number of positive integers less than are equal to 2076 and divisible by 3 or 4.(b) Find the coefficient of x 4y7 in the expansion of (x-y) 11 .

4 . (a) Use the binomial identities to evaluate the sum 1.2.3+2.3.4+.+(n-2)(n-1)n

(b) A store has 25 flags to hang along the fron t of the store to celebrate a sp ecialoccasion. If there ar e 10 red flags, 5 white flags, 4 yellow flags, and 6 blue flags,how many distinguishable ways can the flags be displayed?

UNIT 6:

1. (a) Solve the recurrence relation un+2 -2u n+1 +u n =3n+5 if u0 =0, u1 =1.

(b) Solve the recurrence relation using generating functiona n 6a n 1 = 0 for n 1 and a 0 = 1

2. Solve the recurrence relation a n a n1 + 10a n 2 = 0 for n 2, a 0=13. Solve the recurrence relation a n 9a n 1 + 26a n 2 24a n 3 = 0 for n 3, a 0 =04. (a) Find the generating function of (n-1) 2.

(b) Solve the difference equation u n-2u n1 =5.(2) n using generating function.

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