MODEL EXAMINATION
DISCRETE MATHEMATICS
Year/Branch/Sem : II /IT , CSE / IV Marks: 100
PART-A (20 X 2 = 40)
1. Obtain PDNF for .
2. Give the converse and the Contra positve of the implication “ If it is raining then I get wet”.
3. Represent using only.
4. Determine the truth value of the following a) If 3+4=12 , then 3+2=6.
b) If 3+3=6 , then 3+4=9.
5. State Simple statement function.
6. Express the statement “ For every x there exist a y such that ” in symbolic form.
7. Give the symbolic form of the statement
“ Every book with a blue cover is a Mathematics book”.
8. Symbolize the expression (i) “ All the world loves a lover”
9. For any sets A , B and C , Prove that .
10. Give an example of a lattice which is modular but not distributive.
11. Give an example of a relation which is symmetric but not reflexive
12. Define Characteristic function.
13. If denotes the characteristic function of the set .Prove that
for all .
14. If has 3 elements and has 2 elements. How many functions are there from to .
15. Define odd and even permutation.
16. Show that is a binary operation on the set of positive integers.
17. Let and where is the set of real numbers. Find where .
18. A semi group homomorphism preserves property of associativity
19. Find all the cosets of the subgroup in with the operation multiplication.
20. Define abelian group and subgroup.
PART-B (Answer any 5) (5 X 12 =60 )
21. a)Find the PDNF and PCNF of the formula
.
b) Show that the following premises are inconsistent:
1. If Jack misses many classes through illness and reads a lot of
books.
2. If Jack fails high school, then he is uneducated.
3. If Jack reads a lot of books, then he is not uneducated.
4. Jack misses many classes through illness & reads a lot of
book
22. a)Using Indirect method of proof , derive from
b) Prove that .
23. a)Show that b)Prove that any chain ‘a’ is modular lattice.
24. a) Show that if L is a distributive lattice then for all
.
b)Let R denote a relation on the set of ordered pairs of positive
integers such that iff . Show that R is an
equivalence relation.
25. a)If & are permutations,
prove that .
b) Let the function and be defined and
.Determine the composition function and
.
26. a) Show that .
b) Show that encoding function defined by
is a
group code.
27. State and prove Fundamental theorem on Homomorphism of groups.
28. a) If is the parity check-matrix. Find the Hamming code generated by .If is the received word ,find the corresponding transmitted code word.
b)Find the minimum distance of the encoding function given by .