Prove that the communicating relation defines an equivalence...

Key Concepts

Communicating Relation

The communicating relation is a binary relation defined on a set (often in the context of Markov chains) where two elements are said to communicate if there is a sequence of transitions (or a pathway) from one element to the other and vice versa. This mutual reachability forms the basis for grouping elements into classes where each element in a class is accessible from any other element, implying they are part of the same behavior or structure.

Equivalence Relation

An equivalence relation is a type of binary relation that must satisfy three properties: reflexivity, symmetry, and transitivity. These properties ensure that every element relates to itself (reflexivity), the relation is bidirectional (symmetry), and it maintains consistency across chains of elements (transitivity). When the communicating relation is shown to adhere to these properties, it partitions the set into distinct equivalence classes.

Reflexivity

Reflexivity is the property that every element is related to itself. In the context of the communicating relation, this means that each state or element is considered to communicate with itself, which is an essential requirement for the relation to be an equivalence relation.

Symmetry

Symmetry ensures that if one element is related to another, then the second element must also be related to the first. For the communicating relation, if an element A can reach element B, then there must exist a reverse path ensuring that B can reach A as well, thereby satisfying the symmetry condition.

Transitivity

Transitivity is the property that if one element is related to a second and the second is related to a third, then the first must be related to the third. In the setting of communicating relations, if an element A communicates with B, and B communicates with C, then A should also communicate with C. This property is crucial for establishing well-defined equivalence classes.

Transcript

00:01 In this question, we have been given to check that whether the given relation is an equivalence relation or not.

00:06 So we can say that relation r can be written as x comma y such that x and y have same parents.

00:16 So we can write it over here and it will look like this.

00:20 Now proceeding further, we can here say that let x belongs to the relation r.

00:28 That is we can here say that x comma x will also belongs to relation r this can be written because x and x are going to have same parents because it is the same person written twice right so in mathematical terms we can say that x and x will have same parents so it will be like x comma x belongs to the relation and hence from this very definition we can say that r is reflexive.

01:00 So now we can here recollect the definition of symmetric relation and we can say that let x comma y belongs to the relation.

01:09 Now from this very information what we can say? we can say it means that x and y have same parents, right? so now proceeding further, we will here right that for relation to be symmetric, y comma x should also belongs to the relation are.

01:27 Now what does it mean if it happens let's say it means that y and x also have the same parents? so this is very very simple to understand that if x comma y is having same parents then definitely y comma x will also have the same parents because it is just writing these same two people in a different order.

01:47 So from here we can say that the relation is symmetric and it can be written over here like this.

01:53 Now proceeding further we can say that let x comma y belongs to the relation r and here we can say y comma z belongs to the relation r so from this information we can say that let us understand it first that x comma sorry x y z three things are there now it is given that x and y are having same parents that is what we mean by x x comma y belongs to the relation r.

02:20 Now, y comma z is also having the same parents.

02:25 And this is what we mean by this very information that is written.

02:29 Right...

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