Mathematical Foundations for Computing, Probability & Statistics Computer Science & Allied Engg. branches-21MATCS41 Set-2 Solved Model Question Paper

Module 1

1.A] Define tautology. Show that {(pVq)^(p→r)^(q→r)} → r is a tautology by constructing the truth table.

1.B] Prove the following using the laws of logic [¬p^(¬q ^ r)] v [(q^r) v (p ^ r] ⇔ r.

1.C] For any two odd integers m and n, show that i) m + n is even ii) mn is odd.

2.A] Define i) open statement ii) Quantifiers

2.B] Write the following argument in symbolic form and then establish the validity:
If A gets the Supervisor’s position and works hard, then he will get a raise.
If he gets a raise, then he will buy a car.
He has not purchased a car.
Therefore he did not get the Supervisor’s position or he did not work hard.

2.C] For the following statements, the universe comprises all non-zero integers. Determine the truth value of each statement.

Module 2

3.A] Let A = {1, 2, 3, 4} and B={1, 2, 3, 4, 5, 6}
i) How many functions are there from A to B? How many of these are one-to-one? How many are onto?
ii) How many functions are there from B to A? How many of these are onto? How many are one-to-one?

3.B] Let 𝐴 = {1, 2, 3, 4, 5} × {1, 2, 3, 4, 5} and define R on A by (x1,y1) R(x2,y2) if x1+y1=x2+y2

i) Verify that R is an equivalence relation on A
ii) Determine the equivalence classes [(1, 3)], [(2.4)] and [(1, 1)].

3.C] Define i) Simple graph ii) Complete graph iii) Sub graph iv) Spanning sub graph v) Induced subgraph vi) Complement of a graph vii) Euler Circuit. Give one example each.

4.A] Draw the Hasse diagram representing the positive divisors of 36.

4.B]

4.C] Define Graph isomorphism. Determine whether the following graphs are isomorphic or not.

Module 3

5.A] Calculate the coefficient of correlation and obtain the lines of regression for the following data:

x	1	2	3	4	5	6	7	8	9
y	9	8	10	12	11	13	14	16	15

5.B] Fit a curve by y=ax^b for the following data.

x	1	2	3	4	5
y	0.5	2	4.5	8	12.5

5.C] Fit a straight line in the least square sense for the following data

x	50	70	100	120
y	12	15	21	25

6.A] The following are the percentage of marks in Mathematics(x) and Statistics (y) of nine students. Calculate the rank correlation coefficient.

x	38	50	42	61	43	55	67	46	72
y	41	64	70	75	44	55	62	56	60

6.B] Fit a parabola y=ax² + bx + c for the data and hence estimate y at x = 6.

x	1	2	3	4	5
y	10	12	13	16	19

6.C] With usual notation, compute means, x̄,Ȳ, and correlation coefficient r from the following lines of regression: y=0.516x+33.73 and x=0.512y+32.52

Module 4

7.A] A random variable 𝑋 has the following probability function: