##### Determine the relation

Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) ∈ R if and only if

a) everyone who has visited Web page a has also visited Web page b.

b)  there are no common links found on both Web page a and Web page b

c) there is at least one common link on Web page a and Web page b.

d) there is a Web page that includes links to both Web page a and Web page b.

Things you need to know

Refer to short notes : http://www.techtud.com/short-notes/basics-relation

Now,

a) everyone who has visited Web page a has also visited Web page b.

(I)  Reflexive:          (a, a) ∈ R

because,  everyone who has visited web-page a has by default visited

(II) Transitive:         refer to the def. of Transitive and example on http://www.techtud.com/short-notes/basics-relation

(III) NOT Symmetric: Because, A/Q        (a,b) ∈ R, but (b,a) ∉ R,

(IV) NOT Antisymmetic : Because, A/Q  (a,b) ∈ R, but (b, a) can belong to R for some of webpages..

What this means ?

Consider WebPage = {1,2,3}

and R = { (1,1), (2,2), (3,3),  (1, 2 ) , (2,1), (1,3) , (3,2)}

This relation is antisymetric despite the fact that (1,2) and (1,2) both ∈ R

(b)  there are no common links found on both Web page a and Web page b

(I) NOT Reflexive : As note the relation says that "there are no common links found on both Web page a and Web page b " , i.e if (a, b) ∈ R,

a, b do not share common links ∴ (a,a) ∉ R, hence NOT Reflexive.

(II) Symmentric : Symmetric says that if (a,b) ∈ R, then (b,a ) ∈, this is True for the given relation.

(III)NOT Transitive:  it says that (a, b) ∧ (b, c) ⇒ (a , c) , but if this holds then web-page a and web-page will have a common link to web-                   page c

(IV) NOT Asymmetric: as it is symmetric

(c) there is at least one common link on Web page a and Web page b.

(I) NOT Reflexive : (a,b) ∈ R, if a and b share at-least one common link, but consiter a web page with no links (e) then (e,e) ∉ R. Therefore              not reflexive

(II) Symmetric  : if (a, b) ∈ R, then a and b share atleast one common link, therefore (b,a) ∈ R. Hence , Symmetric

(III) NOT Transitive:    if (a, b) ∈ and (b, c) ∈ R, then it is not necessary that (a,c) ∈ R. Because, if web-page a and web-page b share

at- least one common link and web-page b and web-page c share at-least one common link then, it is not necessary that web-page a and              web-page  will also share a common link.

(IV) NOT Antisymmetric:  as it is symmetric

(d) there is a Web page that includes links to both Web page a and Web page b.

(I) NOT Reflexive: (a,b) ∈ R, if there exist a web-page (say web-page m) such that m has links for both web-page a and web-page b . Now,

(a, a) need not belong to R, as the could be no web-page pointing to web-page a.

(II) Symmetric: if (a, b) ∈ R, i.e there is a web-page linking to web-page a and web-page b, then (b,a) ∈ R. Hence, Symmetric.

(III) NOT Transitive: If  ( a, b) ∈ R, and (b, c) ∈  then it is not necessary that (a,c)∈ R. If web-page e points to web-page a and web-page b,             web-page f points to web-page b and web-page c then it is not necessary that there exit a web-page which'll point to both web-page a and

web-page c.

(IV) NOT Antisymmetric:  As it is symmetric.