 Hi and welcome to the session. Today we will learn about types of relations. So first of all let us understand what is a relation. Let A and B be two non-empty sets then a relation from A to B is a subset of A cross B. Thus one is a relation from A to B if and only if R is a subset of A cross B. So if the ordered pair belongs to R then this implies A is related to B and we write it as A is related to B. Now let us move on to types of relations. So first we have empty relation, a relation in a set A is called relation if no element of set A is related to any element of A that is R is equal to phi which is a subset of A cross A. Next we have universal relation, a relation R in a set A is called universal relation each element of set A is related to every element that is R is equal to A cross A which is a subset of A cross A. Now both empty relation and universal relation are sometimes called trivial relations. Next is equivalence relation. This plays a significant role in mathematics. So to study equivalence relation we will first of all consider three other types of relations that is reflexive, symmetric and transitive. So a relation in a set A is called reflexive if ordered pair A A belongs to R for every A belonging to set A. Next a relation in a set A is called symmetric the ordered pair A1 A2 belonging to R implies that the ordered pair A2 A1 belongs to R for all A1 A2 belonging to set A. And lastly a relation in a set A is called transitive if the ordered pair A1 A2 belonging to R and the ordered pair A2 A3 belonging to R implies that the ordered pair A1 A3 belongs to R for all A1 A2 A3 belonging to set A. So now we have learnt what are reflexive, symmetric and transitive relations. Now let's define equivalence relation a relation R in a set A is said to be an equivalence relation is reflexive, symmetric, transitive. Let's take one example. Here on the set Z of all integers the relation R is defined as the set of ordered pair A, B such that A-V is divisible by 3. And in this we need to show that R is an equivalence relation. So for that we will show that the relation R is reflexive, symmetric and transitive. So first of all let us show that relation R is reflexive let A belongs to Z. Now A-A is equal to 0 which is divisible by 3. This means A is related to A for all A belonging to set Z. So from this we can say that the relation R is reflexive. Next we will show that the relation R is symmetric. So let A, B belongs to Z such that A is related to B. So this implies A-B is divisible by 3 that means minus of A-B is also divisible by 3. From this we get B-A is divisible by 3 that means B is related to A for all A, B belonging to set Z. Thus we get that the relation R is symmetric. Lastly we will show that the relation R is transitive. So for that let us assume that A, B and C belongs to set Z such that A is related to B and B is related to C. Now A is related to B implies that A-B is divisible by 3 and B is related to C implies that B-C is divisible by 3. Now we know that if two numbers are divisible by 3 then their sum will also be divisible by 3 therefore we get A-B plus B-C is divisible by 3. So this implies A-C is divisible by 3 that is A is related to C for all A, B, C belonging to set Z. So from this we get that relation R is transitive. Now we have shown that relation R is reflexive, symmetric and transitive. So that means the relation R is a equivalence relation. Now let us move on to our next topic equivalence class given an arbitrary equivalence relation in an arbitrary set X where R divides into mutually disjoint subsets A i called partitions subdivisions of X satisfying the following conditions. All elements of a particular subset are related to each other that is all elements of A i are related to each other for all i. Second element of a subset is related to any element of some other subset that is no element of A i is related to any element of A j where i is not equal to j and the third condition is union of all the subsets make the whole set X that is union A i is equal to X and all the subsets are disjoint that is A i intersection A j is equal to phi for i not equal to j. So here the subsets are called equivalence classes. Now let us take the same example that is on the set Z the relation R is defined as the set of ordered pairs A B such that A minus B is divisible by 3. For this we have already proved that R is an equivalence relation for this consider the subsets A 1, A 2, A 3 of Z where A 1 is defined by the set X belonging to Z such that X minus 0 is divisible by 3. So in this we will have the integers like minus 6, minus 3, 0, 3, 6, 9 and so on. Now A 2 is defined by the set X belonging to Z such that X minus 1 is divisible by 3. So in this we will have the integers like minus 5, minus 2, 1, 4, 7, 10 and so on and lastly A 3 is defined as the set X belonging to Z such that X minus 2 is divisible by 3. So in this we will have the integers like minus 4, minus 1, 2, 5, 8, 11 and so on. Clearly we can see that all the elements of A 1 are related to each other similarly for A 2 and A 3 and no element of A 1 is related to any element of A 2 or A 3 same is the case for A 2 and A 3. Also A 1 union A 2 union A 3 is equal to Z and A 1, A 2 and A 3 are all mutually disjoint. Now A 1 consites with a set of all integers in Z which are related to 0, A 2 consites with a set of all integers in Z which are related to 1 and A 3 consites with a set of all integers in Z which are related to 2. Thus A 1 is equal to the equivalence class 0, A 2 is equal to the equivalence class 1 and A 3 is equal to the equivalence class 2 and so we have partitioned Z into mutually disjoint equivalence classes that is A 1, A 2 and A 3. Thus in this session we have learnt different types of relations and with this we finish this session. Hope you must have understood all the concepts. Goodbye, take care and have a nice day.