LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS
FOURTH SEMESTER – APRIL 2012
MT 4961 – THEORY OF FUZZY SUBSETS
Date : 25-04-2012 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
Answer all the questions. Each question carries 20 marks.
- a)1) Find the following index of fuzzinessfor the given fuzzy subsets
={(x1/0),(x2/0.3), (x3/0.7),(x4/1), (x5/0), (x6/0.2),(x7/0.6)} and
={(x1/0.3),(x2/1), (x3/0.5),(x4/0.8), (x5/1), (x6/0.5),(x7/0.6)}.
OR
a)2) Give the ordinary subset of level α for the fuzzy subset ={(x1/0.7),(x2/0.5), (x3/1),(x4/0.2), (x5/0.6)} i) α =0.1 ii) α=0.6 iii) α=0.8 iv) 0.9 (5)
b)1) State and prove decomposition theorem for fuzzy subsets. Decompose the fuzzy subset {(x1|0.3), (x2|0.7), (x3|0.5), (x 4|0.1), (x5|0.6)}.
b)2) Let = {(x1/0),(x2/0.3), (x3/0.7),(x4/1), (x5/0), (x6/0.2),(x7/0.6)}
= {(x1/0.3),(x2/1), (x3/0.5),(x4/0.8), (x5/1), (x6/0.5),(x7/0.6)} and
= {(x1/1),(x2/0.5), (x3/0.5),(x4/0.2), (x5/0), (x6/0.2),(x7/0.9)}.
Calculate (6+9)
OR
c)1) Let = {(x1|0.2), (x2|0.7), (x3|1), (x 4|0), (x5|0.5)} and = {(x1|0.5), (x2|0.7), (x3|0), (x 4|0.5), (x5|0.5)}.
Check whether?
c)2) List down all the ‘algebraic’ properties of fuzzy subsets. Explain in detail, giving the implications of those properties that make a difference between crisp sets and fuzzy subsets. (5+10)
- II. a)1) Choosing a suitable example, explain fuzzy subsets induced by a mapping
OR
a)2) Choosing a suitable example, explain normal and global projections. (5)
b)1) Let then prove that ; where is the strongest path existing from x to y of length k.
b)2) Define the algebraic product and sum of two fuzzy relations. Explain with examples. (5+10)
OR
c)1) Prove that the transitive closure of any fuzzy binary relation is transitive binary relation.
c)2) For , and as given below, verify the following conditions.
(5+10)
| A | B | C | D | E | A | B | C | D | E | A | B | C | D | E | |||||
| A | 0 | 1 | 1 | 1 | 1 | A | 1 | 0.5 | 0.5 | 1 | 0.7 | A | 0 | 0.3 | 1 | 0 | 0.5 | ||
| B | 0 | 0 | 0.9 | 0.7 | 0.3 | B | 0 | 1 | 0.7 | 0.7 | 0 | B | 0.3 | 0.2 | 0 | 0.8 | 0.1 | ||
| C | 0 | 0 | 0 | 0.7 | 0.3 | C | 0 | 1 | 1 | 0.7 | 0 | C | 1 | 0 | 0 | 0.2 | 1 | ||
| D | 0 | 0 | 0 | 0 | 0.3 | D | 0 | 0.3 | 0.3 | 0 | 0 | D | 0 | 0.8 | 0.2 | 1 | 0.4 | ||
| E | 0 | 0 | 0 | 0 | 0 | E | 1 | 0.5 | 0 | 0.5 | 1 | E | 0.5 | 0.1 | 1 | 0.4 | 0.4 |
- a)1) Contrast fuzzy ordinal relation with fuzzy resemblance relation. Give an example.
OR
a)2) Consider the relation given with the membership function
Is this relation a resemblance relation? (5)
b)1) Define anti symmetric and perfect anti symmetric fuzzy binary relations. Give examples. Is it true to say that any perfect anti symmetric relation is evidently anti symmetric?
b)2) Let be a similitude relation. Let x, y, z be the elements of E. Put then prove that (7+8)
OR
b)3) Explain the following in detail with examples: Relation of (i)preorder (ii) anti reflexive preorder (iii) similitude (iv)dissimilitude and (v)Resemblance. (15)
- a)1) Explain the three fundamental problems in the process of pattern recognition.
OR
a)2) State the fuzzy c-means algorithm as given J. Bezdek. (5)
b)1) How will you justify that fuzzy applications will yield better results in the field of pattern recognition rather than any other traditional methods.
b)2) Explain how fuzzy clustering methods are based on fuzzy equivalence relation. Given any relation, how is it possible to apply this method. (7+8)
OR
c)1) Explain in detail with examples (i)the two fuzzy clustering methods and (ii) the two pattern recognition methods. (15)
- a) Explain how fuzzy application could make a difference in the field of Economics OR Engineering. (5)
- b) Explain in detail, with a suitable example, fuzzy application in the field of Medicine OR (15)