 Today we will continue our discussion on relations we will see that the relation on a set on a set from from a set to itself that we have defined have different particular properties and we can classify relations based on these properties now before going into all the let us look at let us recall the definition of a relation relation by relation from a set a to a set B we basically mean a subset of the Cartesian product of a subset of the Cartesian product of a by B so any and any subset of a x B that is the Cartesian product of a and B is said to be a relation usually when we say that there is an element a belonging to small a element a belonging to capital A and element B belonging to capital B they are related if by by the relation are if the ordered pair a B belongs to R and we write very often a R B now we go to particular properties of relations the first property in this context is called the reflexive property we say that a relation R on a set a again to recall that when we say that a relation R is on the set on a set a that means the relation R is from the set a to the set a that is R is a subset of the Cartesian product a x a now if R is a relation on a we call R a reflexive relation if a, a belongs to R that is a is related to R for all a belonging to the set a let us try to check one example in this context let us see that let us consider R the set z the set of interiors and consider the relation a R B if and only if a is less than or equal to B so if a is less than or equal to B where a B are elements of the set of integers will say that a is related to B now we see that given any given any a belonging to z a is less than or equal to a in other words a is related to a and this happens for all a in the set of integers therefore less than or equal to relation is a reflexive relation now let us try to think about a relation which is not reflexive so let us let us take the same set z and let us say that for a B belonging to z a is related to B if and only if a into B is strictly greater than 0 please note that here we are specifying that a into B is strictly greater than 0 so it cannot be 0 now let us consider the whether a is related to itself so take any a belonging to z and consider the product of a with itself that is a square now a square is greater than 0 if and only if a is not equal to 0 so a is related to a for all a belonging to z except a not equal to 0 but when a equal to 0 then a 0 is not related to 0 since 0 into 0 equal to 0 is not strictly greater than 0 therefore we see that this relation is not reflexive because there is just one element in the set which is not related to itself therefore we have to be very careful when we are checking for reflexivity of a relation we have to we have to check for each and every relation the each and every element that that will that element should be related to itself now let us check the next property now we say that what happens if a relation is not reflexive so if a relation is not reflexive then well it is it is something that it is possible that a is not related to a for at least some a inside the set on which the relation is defined but if it so happens that a is not related to itself for all a belonging to a then we say that the relation is reflexive that is the second property that we are considering here that is reflexive if a a not in R that is a is not in R for all a belonging to a now let us try to find out one example we can modify the relation that we were looking at again we check yes we say that we define a relation let us say R okay and R is a relation on Z the set of integer defined by x related to y if and only if x is strictly less than y where x, y belongs to Z now take any x belonging to Z of course x is not strictly less than x therefore this relation is reflexive the next in line is the relation the properties symmetric we say that a relation is symmetric if a b belonging to R implies b a belongs to R for all a b belonging to a that is we consider the elements of a and if a for elements a b belonging to a and if it so happens that a is related to b implies always b is related to a then we say that the relation is symmetric this is because this means that this when we are said there is a symmetry about the relation so suppose a related to b if we switch b and a the statement will still be true if the relation is symmetric. So let us look at example of symmetric relation now we have studied congruence modulo m relation so x is congruent to y mod m if and only a m divides y-x this is the congruence modulo relation where x y belongs to Z and m belongs to Z plus now we see that if x is congruent to y mod m then m divides y-x which implies m divides x-y which in turn implies y is congruent to x mod m therefore congruence modulo m relation is a symmetric relation on the set of integers we move on to another property which is particularly important that is called anti-symmetric property now a relation is said to be anti-symmetric if a related to b and b related to a at the same time will mean a equal to b now let us look at an example of anti-symmetric relation we again consider the set z and define a relation that is less than or equal to so the relation is defined as a less than or equal to b we know that is we will say that a less than or equal to b if b-a is greater than 0 now suppose we have a less than or equal to b b less than or equal to a then of course a is equal to b thus this is a anti-symmetric relation the last property that we will study for the time being is called transitive relation so it is like this that if a is related to b and b is related to c then a is related to c let us look at examples of transitive relation we can use the relation that we have already studied that is a less than or equal to b so we see that if a is less than or equal to b and b is less than or equal to c this implies a is less than or equal to c thus it is a transitive relation we will see other examples of transitive relation soon now let us check some more some examples now we are considering z to be the set of integers and abc are elements of z there is a standard notation to define the equivalent relation that is this inverted delta sign so a inverted delta this is a delta right so this delta sign a delta b if and only if a is equal to b so this is the equality relation we can easily see that the equality relation is reflexive because a is always equal to a for all a belonging to z and then a it is symmetric because if a is equal to b then of course b is equal to a it is transitive because if a is equal to b and b is equal to c then a is equal to c and in fact it is also anti-symmetric anti-symmetric because if a equal to b and b equal to a of course a is equal to if a equal to b and b equal to a of course this means that a equal to b but of course we know that it is a equal to b the next relation is less than or equal to relation that we have already seen a less than equal to b if and only if a is less than equal to b and then we have the division a vertical a vertical line b if and only if a divides b we will be reading this symbol as a divides b and then another relation that we have already seen that is a is related to b if and only if a b is strictly greater than 0 and also lastly we have also seen the congruent modulo m relation now we come to the set containment relation suppose S is a set and PS is the power set of S and we will say we will define the relation which be denoted by this this sign if for a b belonging to the power set a contains b if and only if a is a subset of b this is a well known relation this relation is reflexive anti-symmetric and transitive we will very soon see that this relation makes up an important class of relation called partial order relation now we take up some particular type of relations the first one that is considered is called equivalence relation a relation are on a set a is said to be an equivalence relation if and only if the following property is a satisfied x related to x for all x belonging to a that is R is reflexive x is related to y then y is related to x for all x y belonging to a that is R is symmetric if x is related to y and y is related to x then x is related to z for all x y z belonging to a that is R is transitive in other words if we have a relation which is reflexive symmetric and transitive then we call it an equivalence relation possibly the most famous equivalence relation is the congruence modulo m relation that we have studied in the previous lecture and we have considered the example just a while back so we are considering the set of integers and we say that x is congruent to y mod m if y-x is divisible by m for example if we take m equal to 5 then we see that 0 is congruent to 5 mod 5 6 is congruent to 11 mod 5 – 7 is congruent to 3 mod 5 now let us move on to another relation this is we consider the set of real numbers and we say we consider the Cartesian product of the set of real numbers so we are now in a in the real plane so our points are ordered pairs we say that an ordered pair AB in R cross R is related by the relation S to an ordered pair CD in R cross R if and only if a square plus B square is equal to C square plus D square now let us start checking what happens see we are considering R cross R and for two elements AB and CD belonging to R cross R we are defining them to be related by a relation relation let me so they are defining by the relation S AB SCD if a square plus B square equal to C square plus D square what does it mean if we consider the plane R cross R then we will see that suppose let us consider the point 11 here then if we draw a circle around the origin O containing 11 then all the elements that are in the circle that are on the circle has the same distance as that of 11 the distance D here is 1 square plus 1 square it is root 2 so any element for any element on the circle let us say x y x square plus y square has to be equal to 1 square that is equal to 2 so if we consider any other point let us say AB the all the elements related to AB will lie on the circle around the origin O now let us try start checking the properties we claim that given any AB belonging to R cross R a square plus B square of course trivially is equal to a square plus B square which means that AB is S of AB therefore the relation S is reflexive next suppose a, B and C, D both belong to R cross R and a, B SC, D that they are related which implies a square plus B square is equal to C square plus D square this in turn implies C square plus D square is equal to a square plus B square which means that CD is related by S to AB so the relation S is symmetric now let us consider three elements AB CD and let us say EF all elements of R cross R AB related to CD and CD related to EF this together will mean that a square plus B square is equal to C square plus D square C square plus D square is equal to E square plus F square combining these two equations we get a square plus B square equal to E square plus F square this will mean that a, B is related to E, F just let us go back we had here a square plus B square equal to E square plus F square that means you now have AB is related to EF this means that the relation S is symmetric thus S is an equivalence relation on R cross there is another concept which is intrinsically connected to the idea of equivalence relation is the concept of equivalence classes now if R is an equivalence relation on a set are a we choose any X belonging to a and then we consider a set which consists of elements of a which are related to X so let us try to visualize that we have a relation on a set a and we are picking up one element from that set set a let us call it X and we are considering all the all the elements in a which are related to X and we are building up a set which we will call the equivalence class corresponding to X now in this way we can keep on building equivalence classes of each and every element of a then we start questioning that how what is the relationship between different equivalence classes and then we come to a to an observation that if we pick up two elements from a set a then the corresponding equivalence classes are either equal or they are disjoint now this needs a proof so we will check how to prove this fact so let us let us recall X is a equivalence class corresponding to X so it consists of all Y belonging to a on which X is defined such that Y related to X now suppose we consider two elements let us say small a and small b belonging to a and build their equivalence classes which is square bracket a and square bracket b now let us suppose that they have a non-empty intersection that is to say that the intersection of the equivalence class corresponding to a and the equivalence class corresponding to b is not empty therefore there exists a C belonging to a such that C belongs to a intersection b now this in turn means that C is related to a and C is related to C is related to b now this implies that a is related to C and C is related to b since R being an equivalence relation is symmetric therefore I can switch switch a and C they are symmetric C related to a means a related to C and now we will use the transitivity of R because R after all is an equivalence relation and write that this fact implies a related to b but if a is related to b this will imply that a belongs to the equivalence class of b so we see that if the equivalence class corresponding to a and the class corresponding to b have a non-empty intersection then a is an element of the equivalence class of b but then let us consider any element in the equivalence class of a suppose x is an element in the equivalence class of a which implies that x is related to a and since we already know that a is related to b and we also know that R being an equivalence relation is transitive we use the transitive property over here when we write x is related to b and x is related to b means x is inside the congruence the equivalence class of b and therefore this whole thing together implies that the class corresponding to a is contained in the class corresponding to b let us call it one now we could just change the symbols a and b and then everything else will hold true and in exactly the similar way we can prove thus class corresponding to b will be contained in class corresponding to a and combining 1 and 2 we have a equal to b thus we see that if a and b the class corresponding to a and b has non-empty intersection then they are equal so we come to our conclusion that these equivalence classes are either equal or disjoint fact that we prove is that the set of equivalence classes corresponding to an equivalence relation are on a set a partition the set for that we have to first recall what we mean by partition a partition on a set is a set of subsets of that set which are mutually disjoint and which covers the whole set that is for example if we consider if we consider let us look at the so let us consider a set a and p consisting of let us say some subsets of a where this I varies over some indexing set I will say we say that p is a partition of a if and only if di intersection dj is empty for ij belonging to i i not equal to j and union of di when i varies over i gives us the whole set a so suppose r is an equivalence relation on a consider the equivalence classes so consider all the disjoint equivalence classes so suppose you consider this set and we are only considering since it is a set only the no element repeat so we have this a set of subsets which are distinct and we know that distinct equivalence classes are disjoint we claim that this gives us a partition because if we have any if we take any y belonging to a then y is related to y which implies that y is in the equivalence class of y itself therefore for some for therefore this set y must appear in the set so this shows that if I consider the union of all the equivalence classes where x varies over all the disjoint equivalence classes then I will get the whole set a further if we have already proved that if we take any two equivalence classes either they are same in which case this x equivalence class corresponding to x and equivalent class corresponding to y will give us only one entry over here otherwise they are disjoint so they will they will they will contribute to different entries and therefore if we take all the disjoint equivalence classes we will get a partition on the other hand if we are given a partition on a set a then it naturally gives us an equivalence relation let us look at the partition that we have already discussed we can define a equivalent relation in this way that suppose we consider x, y belonging to a we may say that x related to y if and only if x, y belonging to bi now we see that x, x belongs to di because of course x is a single element so this is trivially true so r is reflexive we again see that if x belongs to x and y belongs to di then both things this will imply both things that x, y and y are x therefore x related to y means this which in turn of course means this therefore we know that it is symmetric and further at the end if x related to y and y related to z this implies x and y belongs to di for some i and y, y and z belongs to dj for some j but that means that let us see aside that means that y belongs to di intersection dj we know that di intersection dj is ? if i not equal to j which implies that i equal to j so we can say that this implies that x, y, z belongs to di which means that x is related to z therefore we have transitivity over here at this point I will again emphasize that given any x we will have x belonging to some di because union i, i varying over i di is the whole set a therefore this happens this is the property of partition that we are using thus we have given a proof of the fact that if we have a set and an equivalence relation on a set it will generate a partition on the set and if we have a partition then that it will generate a unequivalence now let us look at some examples of equivalence classes now we go back to the example that we saw some time back which is here here we take m equal to 5 and we have congruence modulo 5 relation and we come over here now corresponding to congruent modulo 5 if we consider 0 let us see the equivalence class generated by 0 so we have got 0 5-5 10-10 15-15 20-20 and so on now if you want to find out the congruence class equivalence class related to 1 then we see that it is 1 then so instead of 0 it is 1 instead of 5 it is 6 instead of 10 it is 11 and so on so this whole equivalence class that we have got corresponding to 0 we have to shift by 1 and we get this one shift by 2 we get this one shift by 3 we get this one and 4 and now we know that if we shift this by another one so if we add one to all the elements we will go back to the congruence class corresponding to congruence or equivalence class corresponding to 0 so these are the equivalence classes corresponding to the congruence modulo 5 relation these are also sometimes called congruence classes now we look at the other other relation that we considered so the equivalence classes here are concentric circles around the origin so we will have infinite number of concentric circles around origin which are the equivalence classes corresponding to the equivalence relation S which is to recall that a b, cd are related by S if a bs square plus b square equal to c square plus d square so this is the end of this lecture so thank you.