Question

Let R be the relation on the set of all people who have visited a particular Web page such that x R y if and only if person x and person y have followed the same set of links starting at this Web page (going from Web page to Web page until they stop using the Web). Show that R is an equivalence relation(i.e. it is reflexive, symmetric and transitive).

Answer 1

Relation R on the set of people who have visited a webpage is an equivalence relation because it is reflexive (each person is related to themselves), symmetric (if x is related to y, then y is related to x), and transitive (if x is related to y and y is related to z, then x is related to z).

Step-by-step explanation:

To demonstrate that relation R is an equivalence relation on the set of all people who have visited a particular webpage we need to show that it satisfies three properties: being reflexive, symmetric, and transitive.

Reflexive

Reflexivity means that every element is related to itself. For a person x, they have followed a set of links starting from the webpage, thus x has followed the same set of links as x itself, which satisfies x R x.

Symmetric

For symmetry, if x R y, then person x and person y have followed the same set of links, and thus y has also followed the same set of links as x, which implies y R x.

Transitive

Lastly, transitivity requires that if x R y and y R z, then x and z have followed the same set of links that y did, resulting in x and z necessarily having followed the same set of links, so x R z.

Since relation R fulfills all three properties, we conclude that R is indeed an equivalence relation.