S2 BCA Module 2 mathematics kerala university

Module 2

Section A

1.      If f: Z→Z where Z is set of integers is defined as f(x) = x+4 write f^-1(x)

2.      How many relations are there on the set {a,b,c,d} ?

3.      If x=<1,0,0,1> and y = <0,1,0,0> then what is the hamming distance H(x,y) ?

4.      Define  equivalence relation

5.      When will you say that a function is invertible?

6.      On the set of rational numbers a relation R is defined as xRy if 1+xy > 0 show that R is reflexive

7.      Define characteristic function

8.      Let A={0,2,4,6,8} B={0,1,2,3,4,5,6} and C={4,5,6,7,8,9,10} find (A∩B)UC

9.      Give an example of a one-one function on A={1,2,3}

10.   Find the inverse F(x) = 2x – 3

11.   Find inverse G(x) =

Section B (answer any ten)

12.   Represent the relation R on a finite set A = {1,2,3,4} defined by aRb if and only if a<b

13.   The function f:Z→Z defined a f(x) = x² + x+1 where Z is set of integers is one-one function prove or disprove.

14.   Find the roots of f(x) over Z₁₀ where f(x) = 2x² +4x+4

15.   Prove that the operation U on sets is commutative

16.   Is the following function one-one?

F: N to N where f(n) = 2n if n is even

                                             n, if n is odd

17.   Let f(x) = x+3 , g(x) = x-4, h(x) = 5x are functions from R to R where R is the set of real numbers. Show that fo(goh) = (fog)oh.

18.   Give examples of sets A and B each having 3 elements such that Ax B and B x A are equal

19.   Give example of a transitive relation which is not symmetric

20.   Define a partial order relation. Give one example specifying the set

21.   Draw the hasse diagram of (D₄₅, /)

22.   Obtain graphical representation of the function f(x)= x²-2x-3

23.   Show that A ∩ (B UC) = (A∩B) U (A∩C)

Section C(answer all)

24.   Let A={1,2,3,4} and a relation R be defined on A on A as R = {(x,y):x>y} draw the graph of R and also give its matrix M_R

25.   Define a fuzzy set. Explain the union and intersection of two fuzzy sets

26.   Define hamming distance and state its properties

27.   Let A be the set of non zero integers and let ~ be the relation on A x A defined by (a,b) ~ (c,d) whenever ad=bc . Check whether ~ is an equivalence relation

28.   Define bijection. Show that if f and g are bijections from X to Y and Y to Z respectively then gof : X to Z is also a bijection

29.   Let A = {1,2,3,4} and R be the relation {(1,2) , (2,3),(3,4),(2,1)} find the transitive closure by using Warshall’s algorithm

30.   Let R= {(1,1),(1,2),(2,3),(3,1)(4,2)} and S={(3,4),(2,1),(3,3),(4,1)} be relations on a set X={1,2,3,4}. write the composite relations RoS, SoR and RoR

Section D(answer all)

31.   Prove that equality modulo m is an equivalence relation

32.   Let R be an equivalence relation on a set A. Prove that R induces a partition on A

33.   State and prove the inclusion exclusion theorem in set theory

34.   Prove that a hamming code can correct all combinations of k or fewer errors if and only if the minimum distance between any two codes is atleast 2k+1

35.   Explain the communication model and error correction in detail

36.   On the set of rational numbers, a relation R is defined by xRy if xy> 0 Check the conditions reflexive, symmetric and transitive for R. is it an equivalence relation? If R does not satisfy the above three conditions modify the definition of R so as to make the condition to be satisfied

37.   Let A,B,C,D be non-empty and different subsets of R. prove or disprove the statement

(A-C) x (B-D) = (A xB) – (CxD)