hey everyone welcome to fussy logic lectures in this video we will be discussing about what is fussy logic introduction to classical sets and fussy sets and extension of classical sets finally we will see some solved examples on the topic so let us start our lecture [Music] what is fussy logic the term logic was introduced by lothfie sade in 1965 with the proposal of fussy set theory the term fussy refers to things which are not clear or are vague in the real world many times we encounter situations where we can't determine whether the state is true or false in such cases fussy logic provides a very valuable flexibility for reasoning so it resembles human reasoning in many ways and hence we can consider uncertainties or inaccuracies of any situation now before we dive deeper into this lecture let us understand what is crisp value the term crisp means precise and it deals with values that has a strict boundary that is true or false the value should be either true or false to be called as crisp value and it cannot contain any in-between values coming to boolean system truth values 1 represents absolute truth value and 0 represents absolute false value that is 1 is true and 0 is false however in the case of fussy logic systems we have intermediate values present which are partially true and partially false now let us consider an example to show how boolean system truth value works and how a fussy logic system works in this question you can see that the condition is is the car moving fast in the boolean system having crisp values you can see two solutions yes and no but in a fussy logic system with fussy values you can see that there are many solutions the car can be moving extremely fast very fast fast slow or extremely slow so you can say that in a boolean system having crisp values you only have absolute true value and absolute false value but in a fussy logic system you have this partially true and partially false values as you can see extremely fast very fast and fast are somewhat close to the absolute true value so you can say that these are partially true similarly slow and extremely slow are somewhat close to the absolute false value therefore you can say that these are partially false so that is the basic difference between fussy logic systems and boolean systems now we will talk about membership functions membership functions were first introduced in 1965 by lothv sade in his first research paper for c sets they characterize fussiness that is all the information in a fussy set no matter whether the elements in the fussy sets are discrete or whether they are continuous so essentially they represent the degree of truth in a system for instance in the previous example for a boolean system having crisp values the solution no represents absolute false value therefore the membership value over here is zero whereas in the case of true value which is yes the membership value is one coming to the case of fussy logic systems it is somewhat different since we are dealing with partially true and partially false values we may have membership values like 0.9 0.8 0.6 0.3 and 0.1 now these values are assigned based on the degree of truth in the fussy input with respect to the first c sets as you can see that extremely fast is more close to absolute true value so i have assigned it a value of power 9 which is close to 1. in similar fashion extremely slow is close to the absolute false value therefore i have assigned it a value of point one which is close to zero so this is how fussy logic systems take up partial values and for each partial output they will be having a membership value assigned to it therefore you can say that fussy logic systems are governed by membership functions let us now move on to introduction to classical sets a classical set is a collection of distinct objects and it has crisp values for example a set of marks of students with marks above 75 so if you write this as a which is a classical set then a can have values such as 76 80 90 95 and so on now each individual entity in a set is called a member or an element of the set that is 76 80 90 95 are called as elements or members of the set a therefore you can say that the classical set is defined in such a way that it has two groups members and non-members here all the marks above 75 comes under the members of set a but any mark which is less than or equal to 75 they are called the non-members of the set a so you can say that in the case of classical sets partial membership does not exist there are only members and non-members now let s be a classical set the membership function that can be used to define the set s is given by chi s of ex where chi is used to denote a classical set is equal to 1 when x belongs to s and c 0 when x does not belong to s so this is the definition of the classical set yes in our example set a an element belongs to set a if its value is greater than 75 and the membership value for all elements belonging to set a is 1 that is chi a of 76 is equal to 1 chi a of 80 is equal to 1 chi a of 90 is equal to 1 and so on in similar fashion if we take chi a of 4 t we know that the value 4 t is not an element of set a so chi a of 4 t is equal to 0 because the membership value for non-members is 0. now another important concept that you need to learn is cardinality cardinality of a set is the number of elements of the set and this number is also referred to as cardinal number let us take an example suppose you have a set b and it has elements a b c and d then you can say that the cardinality of set b is 4 why because there are 4 elements in the set therefore in short the total number of elements of a set constitutes cardinality and in this example the number of elements is 4. therefore the cardinal number over here is 4 okay now we will move on to fussy sets and extension of classical sets fussy sets can be considered as an extension and a gross over simplification of the classical sets we can understand fussy sets in the context of set memberships basically it allows partial membership which means that it contains elements that have varying degrees of membership in a set now in the previous example under fussy logic systems we have seen varying degree of membership like 0.9 0.8 0.6 0.3 and 0.1 in various fussy sets so these represents the partial values and they constitute the degree of membership of the input element in each of these fussy sets that is what is meant by this particular line now we can understand the difference between classical sets and fussy sets classical set contain elements that satisfy precise properties of membership while fussy set contain elements that satisfy imprecise properties of membership this property becomes more evident when you see the graphs for fussy set and classical set so let us consider two graphs one for fussy set and one for classical set as you have learned just now the membership value for a classical set is denoted by chi of x and x over here is the element in similar fashion the membership value for a fussy set is denoted by mu of x we will learn about it in a bit now the graph of a fussy set may look like this whereas the graph of a classical set may look like this the graph of a fussy set looks like this because these particular regions over here they represent the partial membership values partial membership values but in the case of classical set we are only allowed to have c row and one therefore you get a graph like this you will understand this better when we discuss examples later in this video coming to the mathematical definition of fussy sets a fussy set a in the universe u can be defined as a set of ordered pairs and it can be represented mathematically as a where this tilde sign denotes that a is a fussy set is equal to ordered pair y mu a of y where y is the element or value and mu a of y is the membership function and u is the universe of discourse an important point to note here is that the value of mu a of y lies in the range of 0 to 1 with 0 and 1 included in our example discussed earlier extremely fast very fast fast slow and extremely slow these are the different fussy sets just like our set a and we can represent these sets on a fussy graph as a function on a speed scale so let us do that okay so i have an x axis which represents speed and a y axis which represents membership value as you can see x axis has speeds marked in kilometer per hour now let us try to map our fussy sets into this graph now everyone agrees that any speed below 10 kilometer per hour is an extremely slow speed right that means all the speeds between 0 and 10 kilometer per hour are members of the set extremely slow okay since everyone are in agreement that these speeds are extremely slow we can give the membership values of these bits as 1 in the set extremely slow so we can write that mu extremely slow of zero kilometer per hour is equal to one that is the membership value of zero kilometer per hour is one in the set extremely slow now similarly mu extremely slow of one kilometer per hour is also one so the membership value of one kilometer per hour is one in the set extremely slow likewise all the speeds up to mu extremely slow 10 kilometer per hour is one therefore the membership curve for the fussy set extremely slow has membership value equal to 1 for all the speeds in the range of 0 to 10 kilometer per hour okay now the question is what about 11 kilometer per hour does that belong to the set extremely slow well 11 kilometer per hour is definitely faster than 10 kilometer per hour but it is still pretty slow right so we say 11 kilometer per hour is not a full member of the fussy set extremely slow but it is a partial member and let's say 90 percent belongs to it so we can write mu extremely slow of 11 kilometer per hour is equal to 0.9 that is the speed 11 kilometer per hour has a membership value of 0.9 in the fussy set extremely slow let us mark it on the graph so pau and iron lies somewhere here right now using similar arguments we can say that 15 kilometer per hour is only a 50 percent member of the set extremely slow so mu extremely slow 15 kilometer per hour is equal to 0.5 and let us mark it on the graph what about 20 kilometer per hour we can all agree that 20 kilometer per hour is not an extremely slow speed anymore so we can write mu extremely slow 20 kilometer per hour is equal to zero that is 20 kilometer per hour does not belong to the set extremely slow therefore our fuss is set extremely slow can be mapped like this and the membership value of all the speeds above 20 kilometer per hour is also zero in the fussy set extremely slow therefore we get a flat line like this okay so let us write this down this blue graph represents the fussy set extremely slow now using the same arguments i am mapping the fussy set slow let us say the speeds in the range of 20 to 30 kilometer per hour are full members and the speeds in the range of 10 to 20 and 30 to 40 are partial members and the membership value is 0 for all other speeds ok by the way i chose this speed range randomly you are free to choose your own range of speeds for the fussy set slow so this represents the set slow now in similar fashion i'll draw the fussy set fast so we have our fussy set fast so this is first next we have very fast this is very fast finally we have extremely fast so extremely fast now if we give a speed say 65 kilometer per hour as an input or fussy system it will give out the membership value of 65 kilometer per hour in each of these sets so based on the curves we defined the membership value of 65 kilometer per hour is 0 in the fussy set extremely slow so mu extremely slow of 65 kilometer per hour is zero we can write that down here also similarly the membership value of 65 kilometer per hour in the first set slow is also zero so mu slow 65 kilometer per hour is also zero but when it comes to the fussy set fast we can see that 65 kilometer per hour is a partial member and its membership value is 0.5 therefore mu fast 65 kilometer per hour is 0.5 here also 0.5 if you check again you can see that 65 kilometer per hour is also a partial member of the fussy set very fast here also the membership value is 0.5 therefore mu very fast of 65 kilometer per hour is 0.5 we can write that down here also finally as you can see 65 kilometer per hour is not a member of the fussy set extremely fast therefore mu extremely fast 65 kilometer per hour is zero here also we can write it as zero okay so i hope you got a general idea of what a fussy set is and what are membership values one important point to remember is that there are many curves used to define fussy sets here we have used trapezoid shaped curves earlier we have seen triangular shaped curves we can also define fussy sets using sigmoid or gaussian functions therefore we should choose the type of curve which suits our application ok now coming to the representation of fussy sets there are two ways to represent them in the case where universe u is discrete and finite a fussy set a can be represented as membership value of y1 in fascist a by y1 plus membership value of y2 in fascist set a by y2 plus membership value of y3 by y 3 plus etc up to membership value of y n by y n where y 1 y 2 y 3 to y n are elements of fussy set a now this can also be represented as sigma i equal to 1 to n membership value of y i by y i an important point to note here is that these plus signs does not mean actual addition they are just a representation symbol for the collection of elements of y same is the case for these symbols they are not actual divisions or fractions okay there is one more way to represent the same fussy set a it can be written as a equal to ordered pair of y1 membership value of y1 in a comma ordered pair y2 membership value of y2 in a etc till y n comma membership value of y n in a okay so these are the two ways to represent a fussy set let us take an example so that this concept is clear suppose we have a fussy set a with four members one two three and four and the corresponding membership values of members point one point 0.1.3 0.6 and 0.9 using the first representation we learned first set a can be written as membership value of element 1 which is 0.1 by element 1 plus membership value of element 2 which is 0.3 by 2 plus 0.6 by 3 plus 0.9 by 4 now using our second way to represent the same fussy set a can be written as 1 0.1 2 0.3 3.6 and 4 0.9 that is as ordered pair of element and its membership value next when it comes to universe u which is continuous and infinite we will replace the summation symbol with the integral symbol that is fussy set a is equal to integral i equal to 1 to n mu a of y i by y i again i would like to emphasize that this integration symbol does not mean actual integration but simply represents the collection of each element in the set now before ending the lecture let us see a small example to show how crisp and fussy set representations differ for the same situation let set be a set named numbers close to zero define fussy and crisp sets for set and show the membership curve in a graph okay so let us first define a fussy set we will use a triangular curve for this which is centered at 0 like this this curve makes sense right as you can see numbers close to 0 has high membership values and the membership value progressively decreases on either side as we move away from 0. now that we have finalized the membership curve let us define the curve mathematically so we can write mu set of ex equal to 0 when x less than minus 1 which is clear in this graph coming to this region the equation of the straight line is y equal to x plus 1 since here y axis is mu of x we can write mu of x equal to x plus 1 for minus 1 less than or equal to x less than 0. similarly for the region between 0 and 1 we have x minus 1 0 less than or equal to x less than 1 and this equal to 0 when x is greater than 1. so this is the membership function for the fussy set set which we have defined here alternatively we can also use gaussian function to define another fussy set for set like this this curve can be mathematically written as mu set of ex equal to e raised to minus x square where x belongs to r so this is another membership function for the set set here you should note that even though both these membership functions are used to characterize the same description numbers close to 0 both of these are different fussy sets so ideally we should use different labels like mu set 1 of x and mu set 2 of x here here also set 1 and set 2 ok another important point to note is that the properties that a fussy set is used to characterize are usually fussy for example numbers close to 0 is not a precise description right therefore we may use different membership functions to characterize the same description but the membership functions defined are not fussy themselves they are precise mathematical functions so you can see that once a fussy property is represented by a membership function nothing is fussy anymore therefore by characterizing the fussy description with a membership function we have actually diversified the fussy description a common misconception many of us have is that fussy set theory tries to fussify the world but what i explained just now shows otherwise fussy sets are used to defusify the world okay now a doubt many of us will be having is how to determine the membership function curves well sadly there is no one way to do it one approach is to talk with experts in the field and use their knowledge to create a membership function another approach is to gather lots of data related to our application and use that data to figure out a crude membership function and then find unit okay so we have defined fussy sets for set now let us define a crisp set for that we can simply define chi set of ex equal to 1 when minus point 2 less than or equal to x less than or equal to 0.2 and is equal to 0 otherwise and the graph will look like this yn2 sorry minus 0.2 here 4 and 2 and 0 for all other values of x here we have defined that any number between minus 0.2 and 0.2 is close to 0 and all other numbers are not this is the drawback of crisp sets they fail to represent the reality of the world that's all for this lecture i hope that all the concepts taught in this lecture are clear to all of you if you have any doubts feel free to ask them in the comments either me or some other viewer will surely help you out if you found this lecture useful please like the video and support us by subscribing to our channel thank you for watching drop early and have a great day [Music]