Discrete Mathematical Structures

Define subset, disjoint sets and complement of a set with an example for each.

If F₀, F₁, F₂ ---- are Fibonacci numbers prove that 

Prove that every finite integral domain is a field.

Prove that the set Z with binary operations and o defined by :

 is a commutative ring with unity, 

State and prove Lagrange's theorem.

i) Define cycelio group

ii) Prove that the group () is cyclic, Find all its generators. 

The parity check matrix for an encoding function  is given by : 

i) Determine the associated generator matrix

ii) Does this code correct all single errors in transmission? 

A binary symmetric channel bas probability P 0.05 of incorrect transmission. If the word c = 011011101 is transmitted what is the probability that i) single error occurs ii) a double error occurs iii) a triple error occurs? 

Let A = (1, 2, 3, 4, 5, 6, 7). R be the equivalence relation on A that induces the partition A = (1, 2) U (3) U (4,5,7) U (6). Find R. 

Draw the Hasse diagram representing the positive divisors of 36.

Let A = {1, 2, 3, 4) f and g be functions from A to A given by:

f = {(1,4) (2, 1) (3, 2) (4,3)}            g = {(1, 2) (2, 3) (3, 4) (4 19) prove that f and g are inverses of each other.

Let A = R B = {x/x is real and }. Is the function f: defined by f(a)- a² an onto function? a one to one function?  

 Find the number of equivalence relations that can be defined on a finite set A with 

If A = {1, 2, 3, 4) and R is a relation on A defined by R = {(1,2)(1,3) (2, 4) (3,2) (3,3)} find R² and R². Draw their digraphs.

Define Cartesian product of two sets. For non empty sets A, B, C prove that

For the sequence {a_n} defined recursively by  prove that 

State the induction principle. Prove by induction that 6^n-2 +7²ⁿ⁺¹ is divisible by 43 for each positive integer n. 

Prove that for all real numbers x and y if 

 Prove by contradiction that if n² is an odd integer then n is odd". 

Test the validity of the following argument:

    All employers pay their employees

    Anil is an employer 

Anil pays his employees. 

Write the open statement "Some straight lines are parallel or all straight lines interest" in symbolic form and find its negation. 

Establish the validity of the argument

Define converse and inverse of a conditional State the converse and inverse of the following statement "If Ram can solve the puzzle then Ram can solve the problem". 

Prove that  is a tautology using the truth table. 

Define conjunction and disjunction with an example.

Three students write an examination. Their chances of passing are  and respectively.Find the probability that :i) all of them pass ii) at least one of them passes and iii) at least two of them pass. 

In a survey of 260 college students the following data were obtained;

64 had taken mathematics course, 94 had taken computer science, 58 had taken business, 26 had taken both mathematics and computer science, 28. Find taken mathematics and business, 22 had taken computer science and business and. 14 had taken all the three types of courses.

i) How many students were surveyed who had taken none of the three types of courses?

ii) How many had taken only computer science course?