The document discusses propositions and fuzzy logic. It defines a proposition as a statement that can be either true or false but not both, and provides examples. It then explains fuzzy linguistic descriptions and fuzzy rule bases. Finally, it describes several methods for defuzzification including the centroid method, center of sums, and mean of maxima.

2. Consider the statements water boils 90◦c and“sky is
blue”.an agreement or disagreement with these
statement is indicated by a “true or false”value
accorded to the statements.while the first statement
takes on a value false the second takes on a value tree.
A statement which is either ‘true’or ‘false’but not both is
called a proposition.a proposition is indicated by upper
case letter such as P,Q,R.
P:water boils at 90◦c
Q:Sky is blue.

3. 10 20 30 40 50 60
1 1 1 1 1 1
1 1 1 1 1 1
Ṙ(x,y)= 1 1 1 1 1 1
1 1 1 1 1 1
0 0 0.8 1 0.6 0
0 0 0.8 1 0.6 0
0.7 0.7 0.7 0.7 0.7 0.7
1 1 1 1 1 1
To deduce rotation is quit slow we make use of the composition of
the composition rule.

4. Ǭs=vĤ◦Ṝ(x,y)
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
=[0 0 0 0 0 0 0.9 1]* 1 1 1 1 1 1
0 0 0.8 1 0.6 0
0 0 0.8 1 0.6 0
0.7 0.7 0.7 0.7 0.7 0.7
1 1 1 1 1 1
S =[1 1 1 1 1 1]

5. Fuzzy linguistic descriotions are forumal representions of
system made through fuzzy IF_THEN rules. they encode
knowledge about a system in statement of the form IF(a set of
conditions)are satisfied then (a set of consequents)can be
inferred.
IF (x1 is Ā1,x2 is Ā2,……,xn is Ān ) then (y1 isḂ1,y2,Ḃ2,…..yn is Ḃn)
Where linguistic variables xi,yi take the value of fuzzy sets Ai Bj
respectively.
A collection of the rulesnreferring to a particular system is
known as a fuzzy rule base. If the consolusion C Tto be draw
from a rule base R is the conjaction of all the individual
consequents of each rule.
C=C1ᵔC2ᵔ……. ᵔCn

6. Uc (y) = min (Uc1(y),Uc2(y),…….,Ucn(y)),ᵿy ₤ y
On the other hand,if the conclusion c to be
drawn from a rule base R is the disjunction of the
individual consequents of each rule,then
C = C1 ᶸC2ᶸC3…. ᶸCn
Where
Uc(y) = max(Uc1(y),Uc2(y),……,Ucn(y)), ᵿy ₤ y
The conversion of a fuzzy set to single
crisp value is called defuzzification and is the reverse
process of fuzzification.
several methods are avilable in the
literature of which we illustarate a few of the widely
used methods,namely centroid method centre of
sums,and mean of maxima.

7. Also know as the center of gravity or the center of
area method , it obtains the centre of area(x*)occupied by the
fuzzy set.it is given by the expression
x*= ∫u(x)x d x
∫ u(x) d x
For a continous membership function, and
x*= ∑ xi.u(xi)
∑ u(xi)
In the centroid method the overlapping area is
counted once whereas in centre of sums, the overlapping area is
counted twice.cos bulids the resultant membership function each
of the contributing fuzzy sets Ā1, Ā2,……….,the defuzzified value
x* is given by

8. x*= ∑ xi . ∑ UĀk (xi)
∑ ∑ UĀk (xi)
Here n is the number of fuzzy sets and N the number of fuzzy variables
COS is actually the most commonly used deification method. It can be
implemented easily and leads to rather fast interference cycles.
One simple way of deffuzzifing the output is to take the crisp
value with the highest degree of membership.in cases with more than
one element having the maximum value,the mean value of the maxima
is taken.the equation of the defuzzified value x* is given by
x* = ∑ xi
|M|
The height of a fuzzy set A, i.e h(A) is the largest membership
grade obtained by any element in that set.

12. Aea(A)show the area of segments of the aggregated
fuzzy set and Ẋ shows the corresponding centroid now.
x*= ∑AẊ
∑A
x*=18.353/3.715
=4.9
Unlike centoid method the overlapping area is counted not
once but twice. Makinng use of the aggregated fuzzy set in the
centre of sums, x* is given by

13. The areas covered by the fuzzy setsÀ1, À2, À3 are given by
½˟0.3 ˟(3+5) , ½˟0.5 ˟(4+2),and ½˟1˟(3+1) respectively.
Since the aggregated fuzzy set shown in is a continuous set
x* the mean of maxima is computed as x*=6.5.
m={x ₤ [6,7]|U(x)=1} and the height of the aggregated fuzzy
set is 1.