welcome back in this video i will discuss what are membership functions and what are the features of membership functions in fuzzy logic with the help of simple examples first we will understand what is a fuzzy set then we will go to membership function for this set may be viewed as an extension or a generalization of a basic crisp set again there is a question comes in front of us like what is a crisp set here then only we can understand the extension or generalization in this case now let us say that we have been given a set a which has four elements one two three four now what actually happens here is each of these elements are either present in the set a or they are not present for example one is present in set a two is present in set a three and four are present in set a when you say that element five element five is not present in this set a so whatever the elements present are listed here they are present in the set a and whatever they are not listed they are not present in the set a So, there are only two possibilities element is present or not present. So, that is what is called as the crisp set. But in fuzzy set, the extension or generalization is given for this concept with the help of something known as partial membership.
That is, for each of these elements, rather than assigning either zero or one that is either it is present or not present, we can assign something known as a partial membership that is let's assume that for element one, we can assign a membership value as 0.6. What is the meaning of this one is 60% of the time the element is present, 40% of the time this element is not present. Similarly, for the second element, let's assume that I will assign point three, the meaning of this one is 30% of the time this element is present and remaining 70% of the time it is not present.
So, that is what is known as the partial membership for each of these elements here. So, this is also known as the degree of membership for each and every element. And this will range between zero and one in this case. Now, the next question comes in front of us is what is membership function?
Membership function is a curve that defines how each point in the input space is mapped using the membership values between the range of 0 and 1 here. That is, let us assume that there are some elements are present in a particular physics set. Now, how each of those elements are mapped with respect to membership value between 0 and 1 that will be shown with the help of curve here and that is known as the membership function let's take an example in this case the membership value is represented on y-axis and the elements are represented on x-axis here and you can notice here this is the curve which represents the mapping of each element against the membership value that is you can see here this is one element where the membership value may be around 0.2 maybe 0.3, 0.5, 0.8 and the membership value is equal to 1 here and the membership value is getting decreased here. For each of the elements present in the fuzzy set, we have represented those elements with the help of curve against what is known as the membership value here.
So, this is what is known as the membership function in fuzzy set. Now, we can define this membership function formally something like this. Let's say that a fuzzy set A in the inverse of discourse X defined something like this. fuzzy set a is represented with a pair one is the element and another one is the membership value of that particular element for each element in that inverse of discourse here where mu x of a is nothing but the membership function of fuzzerset a in this case so in membership function what we do here is we map this input space against the membership space m here so that is what is known as the membership function in the fuzzy set i said earlier the membership value ranges between 0 and 1 here.
Now, coming back to the features of membership functions, there are mainly three features. The first one is known as the core. The core of a membership function for some fuzzy set A is defined as that region of universe that is characterized by the complete membership in fuzzy set A. That is nothing but you can see here, in this membership function, the element starts from here and they will end here. For each of these elements, there is some membership values assigned here.
Now, The core is that part of this membership function where the membership value of that element is equivalent to 1. So that is what is known as the core here. So you can see here from here to here, the membership value of each element is equivalent to what? 1 here. So that is what is known as the core of the membership function in this case.
Now, the second feature of membership function is the support here. The support of a membership function for a further set A is defined as that region of inverse. that is characterized by non-zero membership of that posisside. The meaning of this one is if a particular element's membership value is non-zero that is known as the support here. now if you look at this one let's assume that there is one element here so what is the membership value it will be more than 0 and less than 1 here if there is one more element is present here the elements membership value is more than 0 and less than 1 here so for all those elements whose membership value is greater than 0 they are called as the support in this case so the support will start from here and it will end over here because for all those elements the membership value is greater than 0 in this case so the support starts here and it will end over here let's say that there is a fuzzy set where the support is a single element a width you can say that the membership value is equivalent to one there is only one element and the elements membership value is equivalent to one such a fuzzy set is called as a fuzzy singleton in this case coming back to the last feature of a membership function that is known as boundary the boundary of a membership function for a fuzzy set a is defined as that region of inverse containing the elements that have non-zero but not complete membership that means you can see here if i consider any element from here to here the element is having the membership value greater than 0 but less than 1 here similarly from here to here the element is having the membership value less than 1 and greater than 0 here it may be present here it may be present here or it may be over here But if the element is present here, the membership value is equivalent to what?
So, that's the reason you can see here from here to here, all those elements are called as the boundary. From here to here, whatever the elements are there, they are called as boundary in this case. now that can be shown mathematically something like this if the membership value is greater than 0 and less than 1 that is known as the boundary here now how to calculate this boundary uh there is one simple trick if you know the support and core the support minus core is nothing but you can say that the boundary over here so this is the simplest technique to calculate the elements which are present on the boundary of a membership function here in this video i have discussed what are fuzzy sets and how they are different from the standard griff sets with the help of simple example also i have discussed what are membership functions and how they are defined mathematically and what are the features of membership functions like core support and boundary with the help of examples i hope the concept of fuzzy sets membership functions and the features of membership function is clear if you like the video do like and share with your friends press the subscribe button for more videos press the bell icon for regular updates thank you for watching