 We will with the definition of a set the question is that what is a set although it is easy to ask this question it is not so easy to answer what has been realized that this simple question leads to several more difficult questions in this lecture we are not going to discuss the difficulties which leads to eventually introduction of axiomatic set theory instead of that we will discuss a somewhat working or operational understanding of a set what we note that in mathematics or in several other applications when we talk about things we somehow understand a collection of things which we are interested in for example when we are talking about integers that is 0 plus or minus 1 plus minus 2 plus minus 3 and all that we rarely think of the set of elephants or set of cows or set of students on the other hand when we consider set of students we usually do not consider the set of complex numbers at the same time or we do not consider set of all possible subsets of the set of integers thus there is an idea of universe which we will denote by script u so when we are talking about set of integers then we will simply say that our universe u is the set of integers the z which is the set of integers we may be talking about real numbers then you will be are the set of real numbers or we may be talking of the set of all students in IIT the set of all students registered in the IITs so these are not numbers these are human beings of course we can label them by numbers we can label them by names but there is a trouble that names of two students may be same so possibly we will label them by their enrollment number and the name of the IIT and like that but of course they are not set of integers nor they are set of all subsets of integers so like that there are different scenarios where we have different universes the elements of an elements of a universe is called objects that is the things that make up the universe in case of integers the objects will be integers in case of real numbers the objects will be real numbers in case of the set of students the objects will be individual students here we will use elements and objects synonymously now once we fix a universe then any collection of well-defined objects inside that universe will be called a set provided that the universe is not too large we are not going to discuss the issues that when will the universe be too large and all the other complicated questions we will simply take the examples that I have discussed already and many such examples where this description of a set works nicely so we can even say that if we fix a universe fixing universe the any collection of any collection of well-defined distinct objects is said to be a set now of course this is not a definition it is just giving an idea of what we will mean by a set it is not a definition because the question that what is a collection will will be raised if you call this a definition and then we have to define somehow the collection the main idea over here is that we fix a universe so we fix the type of objects that we are dealing with and within that type of objects we specify certain objects carefully so that when we encounter an object in that universe we are able to say whether the object that we have encountered has that specified property and this collection or ensemble of these objects which within a universe which specifies properties is called a set for example if we consider the set of integers z which is given by 0 plus 1 minus 1 plus 2 minus 2 and so on if we consider the set of integers which are multiples of 2 that is set of even integers we can now we can easily determine given an element in z whether it is even or not so let us call e as the collection of even integers this collection e is a set the reason is that we have fixed the universe and once we have fixed the universe if I say that I am interested in the set of even integers then given an integer I can determine whether it is even or not and then I can conceive of the collection of even integers in case of objects which are not numbers if I consider the set of all students registered in IITs then we can consider the students registered in IIT Roorkee so the students registered in IIT Roorkee forms a set in the universe of all the students in registered in IITs so we have more or less understood what we mean by a set now we move on to subsets if CD are sets a universe you say that C is a subset of D and write C C every element of C is an element of D in addition D contains an element which is not in C then C is called a proper subset of D and this is denoted by C properly contained in D this symbolically will mean that for all sets CD from a universe you if C is a subset of D then for all x in C implies x in D we introduce some more symbols here that is the symbol this and the symbol inverted a this symbol which looks like epsilon means belongs to belonging to or simply in this means for all thus the sentence that I have written here can be rewritten as for all x x belonging to C here we have another symbol for implies which is this one so I write over here this means implies x belonging to D we will be using this symbols very often so it is good to get used to use to these symbols now there is another symbol that is used frequently which is known as this means there exists we will see say that C is a proper subset of D that is C a proper subset of D implies there exists x belonging to D such that x does not belong to the set C the converse is also true that is if there exists x belonging to D such that does not belong to C and of course here C must be a subset or equal of D then C is said to be a proper subset of D now we question that what is the idea of equality of sets that is when do we say that two sets are equal for a given universe you the sets C and D taken from you are said to be equal we write C equal to D if C is a subset of D and D is a subset of C so here we note the chain of arguments first we give an idea of a set and we give the idea of a universe and then once we give the idea of a set then we give the definition of the subset relation that is we take two sets from the universe and then we say that the set the first set is a subset of the second set if all the elements of the first set are elements of the second set so that is subset equal and if it so happens that the second set has some elements or at least one element which is not in the first set then we will say that the first set is a proper subset of the second set and then we say that if we have got two sets C and D of course consisting of the elements of the universe which we have fixed before starting all these discussions then if it so happens that C is a subset of D and D is a subset of C that is all the elements of C are elements of D and all the elements of D are elements of C then we say that the two sets are equal this leads to some results related to subsets which are somewhat easy therefore I will just write the results and leave the proof to the audience so the first theorem states that let A, B, C are subsets of U that is I have fixed U and A, B, C are sets then if A is a subset of B and B is a subset of C then A is a subset of C B if A is a proper subset of B and B is a subset of C then A is a proper subset of C C if A is a subset of B and B is a proper subset of C then A is a proper subset of C B if A is a subset of B and B is a subset of C then A is a subset of C these are extremely straight forward results and as I have said that I leave it for exercise now let us look at some examples suppose we fix our U to something very small that is 1, 2, 3, 4, 5 and consider two sets A 1, 2, 3 and B 3, 4 now we see that A well another set C 1, 2, 3, 4 so we see that A is a subset of C because A consists of 1, 2, 3 and C consists of 1, 2, 3, 4 further we see that 4 is in C but 4 is not in A therefore A is a proper subset of C similarly we see that B is a proper subset of C but if we consider A and B A is 1, 2, 3 and B is 3, 4 then A is not a subset of B and B is not a subset of A next we introduce another very special set that is called the null set so the null set or the empty set is a unique set containing no element as I have told before that when we are when we have fixed the universe then a set in that universe contains some specific elements of the universe or it may as well contain all the elements the only thing that we expect that when we say that A is a set in the universe U then given an element or an object in U I should be able to decide whether that object is inside A or not now if A is whole of U then the decision is easy because if you give me any object then I know by default that it is in A because A contains all the objects of U if A has some objects in U and some not and some objects in U are not in A then also hypothetically we can have some kind of rule or listing by which we should be able to say that whether an object in U is in A or not but this thing when extended to the other extreme where A does not have any object in U then also A is a well defined collection of objects of U because when we take any object in U we know that it is not in A so A contains no object and this is a very special set called the null set or the empty set and it is denoted by ? or just two braces without any anything inside one result related to the empty set which again is very obvious for any universe U let A is a subset of U then ? that is the empty set is a subset of A if A is not equal to ? then ? is a proper subset of A so this basically says that the empty set ? is a subset of any set A and if A itself is not ? then ? is a proper subset of A the reason behind this is that we say that a set C is a subset of another set D subset of D if x belonging to C implies x belonging to belongs to D now as long as this statement is true that is any x belonging to C belongs to D then C is a subset of D now given any pair C and D if we want to show that C is not a subset of D then we have to find out an element x in C which is not in D so that this statement is false now the trouble here is that when C is equal to ? then C has no element therefore we cannot find an element for which x belongs to C and x not belonging to D holds therefore we cannot prove that this is false and therefore we have to take that ? is a subset of D or in the case of the theorem this is a so therefore what we see is that the empty set ? is a subset of any set we will do this theorem one once more when we formally study logic in after some lectures now we will briefly look at the problem of representing a set so one form of representation we have been doing so far rather intuitively is to put the braces around certain objects for example if we have certain names of students like let us say Sudhir, Aman, Umesh, Dinesh then the set containing these students possibly may be denoted by S and written within braces Sudhir, Aman, Umesh, Dinesh of course we understand that these students are coming from some universe that we have fixed before with numbers we have already seen and we have been doing that suppose we want numbers from 1 to 5 that is integers from 1 to 5 we have just written something like this suppose I denoted by n and I write 1, 2, 3, 4, 5 so in this way we can collect the objects and put braces around them and represent a set but as is evident to all of us that if we have an infinite set or a very large set it will be very difficult to write all the elements or more often impossible to write all the elements within braces then we take some other alternatives for example what we can do is that first we fix the universe you suppose I want all integers from 1 to 10000 then fixing the universe you I can write the set let us say I as x belonging to you such that 1 less than or equal to x less than or equal to 10000 thus we can avoid listing down 10000 integers by writing this inequality and understanding that you is a set of integers but we have to be careful here because it depends on how we choose the integer how we choose the universe because suppose we choose the universe to be set of real numbers then instead of a set of integers this same looking set will be an interval which I am writing as int between 1 to 10000 of course it is possible to write other sets in the same way for example suppose we would write we would like to write all the matrices all the 2 by 2 matrices with real entries whose determinant values is non 0 then we may write it like this that g is equal to a b c d such that a, b, c, d belongs to R and ad-bc is not equal to 0 of course this is an infinite set so I cannot write all the 2 by 2 matrices with real entries with non 0 determinant I cannot list them but with this notation I can specify them quite precisely next we move on to the idea of the power set of a set now suppose a is a set of objects from a universe u the power set of a is a set of all subsets of a now let us look at an example suppose a is a finite set containing only three elements now we start writing the set of all subsets of a the first and the most obvious which is a subset of any set is Phi that is a set containing no element and then we have the sets containing just one element that is 1, 2, 3 and then we have the sets containing two elements 1, 2, 1, 3 and 2, 3 and finally we have the set containing three elements that is 1, 2, 3 and if we put braces around all these sets then we have the power set of a which we denote by Pa now if we count the number of elements in Pa we see that this is 1, 2, 3, 4, 5, 6, 7 and 8 we write that as simply writing Pa within two horizontal lines and this is 8 by very easy counting argument we can prove that if we have a set containing n elements then its power set will contain 2 to the power n elements suppose a is a finite set containing n elements then Pa will contain 2 to the power n elements and the notation that we have used here is quite general so if we have a finite set then by writing that set symbol within two vertical lines we denote the number of elements in that set suppose S is a finite set then is the number of elements in S in today's lecture we have started with the definition or not really the definition but a description of the idea of a set we have talked about how to conceive of a universe and then conceive of some the some collections of specific objects within the universe and after that basic description we have defined the terms such as subsets proper subsets null sets null set or empty set and lastly we have talked about power sets and number of elements in a set in the next lectures we will discuss more on sets such as operations on sets and laws of set operations but this is all for today thank you.