 Hello everyone, I am Reshna Pathak from Valchand Institute of Technology, Sholapur. So today in this video session we are going to see all about functions. Now function is nothing but a class of a relation. What will be the learning outcome for this session? At the end of this session student will be able to explain functions in discrete mathematical structures. Now let us see a simple definition of a function. A function takes an element from a set and maps it to a unique element in another set. So this is nothing but your function. Suppose you have a set R and you have another set say Z and you want to map the element from this set to this set then it is possible with the help of your function. And the mapping is named with a particular symbol. So suppose now you have element 4.3 in set R and another element 4 in set Z. So I am mapping this 4.3 with element 4. So the elements present in R are called as domain and the elements present in set Z are called as po domain. And the 4.3 is nothing but it is a pre-image of your 4 and this is nothing but your image of 4.3. So this is all about a function. Now let us see a standard definition for function. Given any two sets say A and B a function named as F from or we say mapping corresponding so these are basically synonyms used for your functions. So given any sets A and B a function F from A to B that is F to B so this symbol is used for function A to B is an assignment of exactly one element that is F of X which belongs to B to each element where your X belongs to A that is capital A uppercase A it means your set. Now this is graphical representation of your function. You see we have a set B and we have a set A another set A so I am mapping element A to element B that is element from set B to the element to set A and this mapping is done with the F. So this is nothing but graphical representation of a function. Now we have some terminologies which we use in our function. Let us see in detail with an example. So if F is A to B that is A is mapped to B and F of A equals to B where your A belongs to your lowercase A belongs to capital A is nothing but your set A and B is nothing but element in set B. So B belongs to capital B then so if this is the case then we say A is domain of F your capital B that is uppercase B is codomain of F your B is the image that means element is the image of A under F and A is a pre-image of B under F. So in general we see that B may have more than one pre-image. So these are nothing but the range where R is subset of B of F is read as B such that there exists A where your F of A equals to B. So these are nothing but your terminology is used in function. Now let us see more in detail about domain and codomain. So I have given a simple example where your A class grade function. See I have some elements over here and I have few elements where we have mapped these elements to these elements. So this is nothing but your domain and your codomain. So we find your domain that means elements at this side are mapped with elements from this that is your codomain. We have one more example that is a string length function. Now here we have A a pre-image of 1. So basically we are doing nothing but the length we are calculating the string length. So for this it is 1 and the image of A is 1. So pre-image of 1 is A the image of A is 1. So this is nothing but your pre-image and image. Now I have a question for you. Suppose that F is a function mapping students in this class to the set of grades A, B, C, D and E. At this point you must say what is codomain and its range. So F's codomain what will be the codomain for your F and range. Now suppose the grade turns all A's and B's then what will be the range of F but what will be the codomain. So take a pause think on it. So the answer is you know F's codomain will be always A, B, C, D and E. Of course its range will be unknown. Now suppose the grades turn all A's and B's then the range of F is obviously your A, B and the codomain will be A, B, C, D and E. So this was all about your domain and codomain. Let us take one more simple example for this. You have say set X and you have another set Y. So the elements in X are nothing but your domain and the elements in Y are your codomain. So I will use a function F to map these elements. So I have 1, 2, 5, 7, P, Q, Jack, Q and in X I have 1, 5, P and Jack. And elements in Y are 2, 5, 7 and gel. Now my question is what will be the domain for F and what will be the range for your function F. So obviously all the elements present in your X element are domain. So the domain of F is always your X. So this is your domain. Now let us see range. We have 1, 5, P and Jack, 1, 5, P and Jack as domain that is nothing but your X coordinate. We have 2, 7, Q and Q. So these are your Y coordinate which will fall under your range of a function. So here I have 2, 7, Q repeated will be avoided. And here you have 2, 7 and Q. So these are nothing but your domain and ranges of function. One more thing, we have F of 1 equals to 2. So this was the case 1, 2. So this is read as function of 1 is nothing but your 2. Suppose if it is Jack, Q. So this will be read as F of Jack will be equal to Q. So this is all about your function. So this is all about your function. Now further we see types of function. In types of function we have basically 3 types. First is your surjective function that is also termed as onto function. Another will be your injective function which is nothing but your 1 to 1 function. And the third one is your bijective function which is combination of your surjective and injective function. So that is your onto and 1 to 1. Now basically in surjective function all the elements from your domain should be mapped with your co-domain in onto manner. It means it is not mandatory to map all the elements. You can leave some unmapped element into your co-domain. Now in injective function that is nothing but your 1 to 1 element. So the mapping will be done from 1 to 1. So all the elements from your domain will be mapped to your co-domain. And in case of your bijective function it is onto and 1 to 1. So the mapping should take place 1 to 1 and it should be onto. So these are nothing but your types of function. Now in this session we have studied all about functions. So here are some of the references which I have used during the content preparation. Thank you.