2015

(November)

MATHEMATICS

(Major)

Course: 501

(Logic and Combinatorics, and Analysis - III)

Full Marks: 80

Pass Marks: 32

Time: 3 hours

The figures in the margin indicate full marks for the questions

(A) Logic and Combinatorics

(Marks: 35)

1. (a) Define a truth function. 1

(b) Let: Ice is cold, : Blood is green. Write the following sentences in symbolic form: 2

- Either ice is cold or blood is green.
- If blood is green then ice is not cold.

(c) Prove that is a tautology. 3

(d) Prove that every statement can be generated by only. 4

Or

Prove that

is a valid statement. 4

2. (a) Write the law of syllogism. 1

(b) Using predicates, write the following sentences in symbolic form: 2

- All teachers like all students.
- Only teachers like students.

(c) Test the validity of the following arguments: 3

All squares have equal sides.

A rhombus has equal sides.

Therefore, a rhombus is a square.

(d) Prove that is a valid consequence of the following premises: 4

Or

All men are mortal.

Ram is a man.

Hence Ram is mortal.

Write the formal derivation.

3. (a) Write the fundamental principles of counting. 1

(b) A computer password consists of a letter of the alphabet followed by 3 or 4 digits. Find the total number of passwords that can be formed. 2

(c) Define Catalan number. Prove that the nth Catalan number defined from to is given by 1+3=4

Or

Define Stirling number of first kind. Find the number of functions from a set of elements to a set of elements such that the ranges of these functions have exactly elements each. 4

4. (a) Write the Pigeonhole theorem. 1

(b) Prove that given any 12 natural numbers, one can choose 2 of them such that their difference is divisible by 11. 3

(c) Find the number of solutions in integers of the equation 4

Or

Use generating functions to find the number of ways to select objects of different kinds if we must select at least one object of each kind. 4

(B) Analysis – III (Complex Analysis)

(Marks: 45)

5. (a) Define continuity of a function of a complex variable. 1

(b) If

Then prove that is not differentiable at. 3

(c) Prove that the function

is not analytic at , although Cauchy-Riemann equations are satisfied at that point. 6

Or

Prove that is harmonic. Find its harmonic conjugate. 6

6. (a) Define Jordan arc. 1

(b) Evaluate along the path 1

(c) State and prove Liouville’s theorem. 5

(d) Answer the following (any one): 4

- Using Cauchy integral formula, evaluate

where is the circle .

- Evaluate, where is given by.

7. (a) Define radius of convergence of a power series. 2

(b) Expand in the region. 3

(c) Expand in a Taylor’s series about 3

Or

Expand for. 3

8. (a) Define isolated singular point. 1

(b) Find the poles of. 2

(c) Evaluate (any two): 5x2=10

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